geometry: a complete course€¦ · lesson 3 - theorem 15 - “if two lines intersect, then the...

Written by: Larry E. Collins

Geometry:A Complete Course

(with Trigonometry)

Module C – Solutions Manual

RobbinsCreative

Errata March 2015

Geometry: A Complete Course (with Trigonometry)Module C–Solutions ManualCopyright © 2014 by VideoTextInteractive

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Table of ContentsUnit III - Fundamental TheoremsPart A - Deductive Proof

LESSON 1 - Direct Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1LESSON 2 - Indirect Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Part B - Theorems About Points and LinesLESSON 1 - Theorem 1 - “If a point lies outside a line, then exactly

one plane contains the line and the point” . . . . . . . . . . . . . . . . . . . . . . . . .5LESSON 2 - Theorem 2 - “If three different points are on a line,

then at most one is between the other two” . . . . . . . . . . . . . . . . . . . . . . . .8

Part C - Theorems About Segments and RaysLESSON 1 - Theorem 3 - “If you have a given ray, then there is exactly one

point at a given distance from the endpoint of the ray” . . . . . . . . . . . . . .11LESSON 2 - Theorem 4 - “if you have a given line segment, then that

segment has exactly on midpoint” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Part D - Theorems About Two LinesLESSON 1 - Theorem 5 - “If two different lines intersect, then exactly one

plane contains both lines.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18LESSON 2 - Theorem 6 - “If in a plane, there is a point on a line, then there

is exactly one perpendicular to the line, through that point.” . . . . . . . . .22

Part E - Theorems About Angles - Part 1 (One Angle)LESSON 1 - Theorem 7 - “If, in a half-plane there is a point on a line, then

there is exactly one other ray through the endpoint of the given ray,such that the angle formed by the two rays has a given measure.” . . . . .26

LESSON 2 - Theorem 8 - “If, in a half-plane, you have an angle, then that angle has exactly one bisector.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

Part F - Theorems About Angles - Part 2 (Two Angles)LESSON 1 - Theorem 9 - “If two adjacent acute angles have their exterior sides

in perpendicular lines, then the two angles are complementary.” . . . . . .34LESSON 2 - Theorem 10 - “If the exterior sides of two adjacent angles are

opposite rays, then the angles are supplementary.” . . . . . . . . . . . . . . . . .36LESSON 3 - Theorem 11 - “If you have right angles, then those right angles

are congruent.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40LESSON 4 - Theorem 12 - “If you have straight angles, then those straight

angles are equal.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

Module C - Table of Contents

Part G - Theorems About Angles - Part 3 (More than Two Angles)LESSON 1 - Theorem 13 - “If two angles are complementary to the same

angle or congruent angles, then they are congruent to each other.” . . . 46LESSON 2 - Theorem 14 - “If two angles are supplementary to the same angle

or congruent angles, then they are congruent to each other.” . . . . . . . . .50 LESSON 3 - Theorem 15 - “If two lines intersect, then the vertical angles

formed are congruent.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53

Part H - Theorems About Parallel LinesLESSON 1 - Postulate 11- Corresponding Angles of Parallel Lines . . . . . . . . . . . . . . .58 LESSON 2 - Theorem 16 - “If two parallel lines are cut by a transversal,

then alternate interior angles are congruent.” . . . . . . . . . . . . . . . . . . . . . .60LESSON 3 - Theorem 17 - “If two parallel lines are cut by a transversal,

then interior angles on the same side of the transversal are supplementary.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

LESSON 4 - Theorem 18 - “If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.” . . . . . . . . . . . . . . . .69

LESSON 5 - Theorem 19 - “If two lines are cut by a transversal so that corresponding angles are congruent, then the two lines are parallel.” . .73

LESSON 6 - Theorem 20 - “If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.” .76

LESSON 7 - Theorem 21 - “If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary,then the two lines are parallel.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

LESSON 8 - Theorem 22 - “If two lines are perpendicular to a third line,then the two lines are parallel.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83

LESSON 9 - Theorem 23 - “If two lines are parallel to a third line, then thetwo lines are parallel to each other.” . . . . . . . . . . . . . . . . . . . . . . . . . . .88

LESSON 10 - Theorem 24 - “If two parallel planes are cut by a third plane, then the two lines of intersection are parallel.” . . . . . . . . . . . . . . . . . . . .89

Module C - Table of Contents

Unit III — Fundamental TheoremsPart A — Deductive Proof

p. 215 – Lesson 1 — Direct Proof

1. a) Reflexive Property of Equality f) Multiplication Property of Equality

b) Transitive Property of Equality g) Distributive Property of Multiplication to Addition

c) Symmetric Property of Equality h) Addition Property of Equality

d) Addition Property for Equations i) Substitution Principle

e) Multiplication Property for Equations j) Property of Zero for Addition

2. b) Addition Property for Equations g) Multiplication Property for Equations

c) Arithmetic Fact h) Associative Property for Multiplication

d) Commutative Property for Addition i) Multiplicative Inverse Property

e) Additive Inverse Property j) Property of One for Multiplication

f) Property of Zero for Addition k) Substitution

3. b) Multiplication Property for Equations e) Property of One for Multiplication

c) Associative Property for Multiplication f) Substitution

d) Multiplicative Inverse Property

4. b) Multiplication Property for Equations f) Addition Property of Equality

c) Multiplicative Inverse Property g) Additive Inverse Property

d) Property of One for Multiplication h) Property of Zero for Addition

e) Substitution i) Substitution

5. 1. Given 4. Postulate 7 - Protractor - Fourth Assumption

2. Given 5. Postulate 7 - Protractor - Fourth Assumption

3. Addition Property for Equality 6. Substitution

6. 2. ST; Postulate 6 - Ruler - Fourth Assumption 6. Addition Property of Equality

3. RN; Postulate 6 - Ruler - Fourth Assumption 7. Additive Inverse Property

4. Substitution Principle 8. Property of Zero for Addition

5. Given

7. 2. Definition of Segment Congruence 6. Postulate 6 - Ruler - Fourth Assumption

3. Reflexive Property of Equality 7. FA = LT

4. Addition Property for Equality 8. Definition of Segment Congruence

5. Postulate 6 - Ruler - Fourth Assumption

8. Conditional: If 2x + 8 = – 16, then x = – 12.

Given: 2x + 8 = – 16 Prove: x = – 12

Statement Reason

1. 2x + 8 = – 16 1. Given

2. 2. Addition Property of Equality.

3. 3. Additive Inverse Property

4. 4. Property of Zero for Addition

5. 2x = – 24 5. Substitution

6. 6. Multiplication Property for Equations

7. 7. Associative Property of Multiplication

8. 8. Multiplicative Inverse Property

1Part A – Deductive Proof

2x + 8 + -8 = -16 + -8

2x + 0 = -16� +� -8

2x = -16� +� -8

12

12

2 = -24!

( )

( )x

12

12

( )

( ) 2 = -24x

1 =12

-24! x

( )

• •

•

2 Unit III – Fundamental Theorems

8.Continued

Statement Reason

9. 9. Property of One for Multiplication

10. x = -12 10. Substitution

9. Conditional: In nRST, if /1 > /3 and /2 > /4, then /SRT > /STR

Given: /1 > /3 Prove: /SRT > /STR

/2 > /4

Statement Reason

1. /1 > /3 1. Given

/2 > /4

2. m/1 = m/3 2. Definition of Congruence

m/2 = m/4

3. m/1 + m/2 = m/3 + m/4 3. Addition Property for Equality

4. m/1 + m/2 = m/SRT 4. Postulate 7 - Protractor - Fourth Assumption

m/3 + m/4 = m/STR

5. m/SRT = m/STR 5. Substitution

6. /SRT > /STR 6. Definition of Congruence

10. Conditional: In nRST, if RQ > TP and MQ > MP, then RM > TM.

Given: RQ > TP Prove: RM > TM

MQ > MP

Statement Reason

1. RQ > TP , MQ > MP 1. Given

2. RQ = TP 2. Definition of Congruence

3. RM + MQ = RQ 3. Postulate 6 - Ruler (Segment–Addition Assumption)

TM + MP = TP

4. RM + MQ = TM + MP 4. Substitution (statements 3 and 1)

5. MQ = MP 5. Definition of Congruence

6. RM + MQ = TM + MP 6. Substitution

7. RM + MQ - MQ = TM + MP -MP 7. Subtraction Property of Equality

8. RM + 0 = TM + 0 8. Additive Inverse Property

9. RM = TM 9. Property of Zero for Addition

10. RM > TM 10. Definition of Congruence

x =12

-24

( )

Unit III — Fundamental TheoremsPart A — Deductive Proof

p. 223 – Lesson 2 — Indirect Proof

1. a) The sun is not shining.

b) MN > PQ

c) /A is acute.

d) nABC is not congruent to nDEF.

2. a) does not necessarily lead to a contradiction

b) does lead to a contradiction

c) does not necessarily lead to a contradiction

d) does not necessarily lead to a contradiction

3. a) Suppose the light bulb is not defective.

b) Suppose the number has more than two integer factors.

c) Suppose the two angles are not congruent.

d) Suppose A, B, and C are not collinear.

e) Suppose the two lines are not skew lines.

4. a) Conditional - If line , passes through point P and is parallel to line k, and line m passes through point P and

is parallel to line k, then line , and line m are the same line.

b) Diagram-

c) Given - , is a line that passes through point P.

m is a line that passes through point P.

, II k and m Il k.

d) Prove: , and m are the same line.

e) Proof:

1. Suppose line , and line m are not the same line. 1. Indirect Proof Assumption

2. Line , passes through point P and is parallel 2. Given

to line k.

3. Line m passes through point P and is parallel 3. Given

to line k.

4. There are two lines through point P parallel 4. Steps 2 and 3

to line k.

5. Step 4 is a contradiction. 5. Unit II - Postulate 9 - In a plane, through a point not on a given

line, there is exactly one line parallel to the given line.

6. Our assumption is false, line , and line m 6. R.A.A.

are the same line.

5. a) Conditional - If /AXY and /YXB form a linear pair and m/AXY > 90, then /YXB is not a right angle.

b) Diagram-

c) Given - /AXY and /YXB form a linear pair; m/AXY > 90

d) Prove - /YXB is not a right angle.

3Part A – Deductive Proof

m

k

P

A X B

Y

>90º

5. - continued

e) Proof

1. Suppose /YXB is a right angle. 1. Indirect Proof Assumption

2. m/YXB = 90O 2. A right angle is an angle whose measure is 90º

3. /AXY and /YXB form a linear pair. 3. Given

4. AX and XB are opposite rays. 4. Definition of Linear Pair

5. /AXB is a straight angle 5. Definition of Straight Angle

6. m/AXB is 180 6. Definition of Straight Angle

7. /AXY and /YXB are supplementary. 7. Defintion of supplementary angles.

8. m/AXY + m/YXB = 180O 8. Definition of supplementary angles

9. m/AXY + 90 = 180 9. Substitution

10. m/AXY + 90 + -90 = 180 + -90 10. Addition Property of Equality

11. m/AXY + 0 = 180 + -90 11. Additive Inverse Property Number Fact

12. m/AXY = 180 + -90 12. Identity Property for Addition

13. m/AXY = 90 13. Substitution

14. However, m/AXY > 90 14. Given

15. Steps 10 and 11 represent a contradiction. m/AXY 15. Postulate 7 - Protractor - Second Assumption - cannot both equal 90 and be greater than 90. an angle has a unique measure

16. Our original assumption must be false. 16. R.A.A.m/YXB > 90 and therefore, not a right angle.

6. a) Conditional: If a triangle is a right triangle, then it has no more than two acute angles.

b) Diagram

c) Given - nABC is a right triangle with right angle C.

d) Prove - nABC has no more than two acute angles

e) Proof

1. Suppose nABC has three acute angles, 1. Indirect Proof Assumption

/A, /B, and /C

2. m/C < 90O 2. Definition of Acute Angle

3. nABC is a right triangle with right angle C. 3. Given

4. m/C = 90O 4. A right angle is an angle whose measure is 90º

5. Steps 2 and 4 represent a contradiction. 5. Postulate 7 - Protractor - Second Assumption -

m/C cannot both equal 90 and be less than 90 measure of an angle is a unique real number.

6. Our original assumption must be false. 6. R.A.A.

nABC is a right triangle with no more than

two acute angles.


A

C B

Unit III — Fundamental TheoremsPart B — Theorems about Points and Lines

p. 228 – Lesson 1 — Theorem 1: If a point lies outside a line, then exactly one plane contains the line and the point.

1. a) If a given point lies outside a given line, then exactly one plane contains the given line and the given point.

b)

c) Given: Point N lies outside line ,d) Prove: Exactly one plane contains point N and line ,.

e)

Statement Reason

1. Point N lies outside line , 1. Given

2. Points A and B are on line , 2. Every line contains at lest two points. (Postulate 1)

3. There is exactly one plane containing 3. For any three different non-collinear points,

points A,B, and N. there is EXACTLY ONE PLANE containing them.(Postulate 2)

4. line , is in plane P 4. For any two different points in a plane, the line

containing them is in the plane.

5. exactly one plane contains the given 5. Q.E.D.

line and the given point.

2.

3. Yes; We know from Theorem1, that point A and line DC are in exactly one plane. (If a point (point A) lies outside a line(Line DC or

Line DB on Line CB), then exactly one plane contains the line and the point.) Since the points D, C, and B are on Line DC, the

points are all in the same plane as the line. Points A, B, C, and D lie in the same plane.

4. Conditional: If nABC, with side BC in BD, has interior /ACB congruent to /ACD, then nABC is a right triangle.

a) I can prove triangle ABC is a right triangle if I can prove triangle ABC has a right angle. I can prove /ACB measures 90O. I can

prove this if I can prove m/ACB is half the measure of a straight angle which measures 180O. The rays, CB and CD are

opposite rays which makes /BCD a straight angle, so I can do this plan. Start at the end of the analysis and work backwards.

b) Statement Reason

1. nABC with side BC on BD 1. Given

2. /BCD is a straight angle. 2. Definition of a Straight Angle -

sides of angle are opposite rays by observation

3. m/BCD = 180 3. The measure of a straight angle is 180º.

4. m/BCA + m/ACD = m/BCD 4. Postulate 7 - Protractor - Fourth Assumption

5. m/ACB + m/ACD = 180 5. Substitution

6. /ACD > /ACB 6. Given

7. m/ACD = m/ACB 7. Definition of Congruent Angles

8. m/BCA + m/ACB = 180° 8. Substitution

9. (1 + 1) • m/ACB = 180° 9. Distributive Property of Multiplication to Addition

The figure appears to be

a right triangle.

N

A

B

B C D

A

5Part B – Theorems About Points and Lines

4. b) continued.

Statement Reason

10. 2 • m/ACB = 180O 10. Substitution

11. 11. Multiplication Property of Equality


13. 1 • m/ACB = 90O 13. Substitution

14. m/ACB = 90 14. Multiplication Property of One

15. /ACB is a right angle 15. Definition of Right Angle

16. nACB is a right angle 16. Definition of Right Triangle

5. a) I can prove /ABD and /CBD are complimentary if I can prove the sum of their measures is 90O. I can prove their sum is 90 if I

can prove m/ABC is 90O. I know /3 is a right angle (90O) and forms a linear pair with /4 (which makes /4

equal to 90O). /4 is congruent to /ABC, so I can do this plan. Start at the end of the analysis and work backwards.

b) Statement Reason

1. /ADC is a straight angle. 1. Definition of Straight Angle- sides of angle are opposite rays

2. m/ADC = 180 2. The measure of a straight angle is 180º.

3. m/3 + m/4 = m/ADC 3. Postulate 7 - Protractor - Fourth Assumption

4. /3 is a right angle 4. Given

5. m/3 = 90 5. The measure of a right angle is 90º

6. 90 + m/4 = 180 6. Substitution of Equality (5 and 2 into 3)

7. 90 + m/4 + -90 = 180 + -90 7. Addition Property for Equations

8. 90 + -90 + m/4 = 180 + -90 8. Commutative Property of Addition

9. 0 + m/4 = 180 + -90 9. Additive Inverse Property

10. m/4 = 180 + -90 10. Property of Zero for Addition

11. m/4 = 90 11. Arithmetic Fact or Substitution

12. /4 > /ABC 12. Given

13. m/4 = m/ABC 13. Definition of Congruent Angles

14. 90 = m/ABC 14. Substitution

15. m/ABD + m/DBC = m/ABC 15. Postulate 7 - Protractor - Fourth Assumption

16. m/ABD + m/DBC = 90 16. Substitution

17. /ABD and /DBC are complementary angles 17. Definition of Complementary Angles

(sum of Two Angle measures is 90 degrees)

1

22 m ACB =

1

2180O⋅ ⋅ ∠ ⋅


1 m ACB = 1

2180O⋅ ∠ ⋅

6. a) Statement Reason

1. AB > CD 1. Given

2. AB = CD 2. Definition of Congruent Segments

3. BC = BC 3. Reflexive Property of Equality

4. AB + BC = BC + CD 4. Additional Property for Equations

5. AB + BC = AC 5. Postulate 6 - Ruler - Fourth Assumption

BC + CD = BD

6. AC = BD 6. Substitution Principle (5 into 4)

7. AC > BD 7. Definition of Congruent Segments

b) Statement Reason

1. AC > BD 1. Given

2. AC = BD 2. Definition of Congruent Segment


BC + CD = BD

4. AB + BC = BC + CD 4. Substitution

5. BC = BC 5. Reflexive Property of Equality

6. AB + BC - BC = BC + CD - BC 6. Subtraction Property of Equality

7. AB + BC - BC = CD + BC - BC 7. Commutative Property of Addition

8. AB + 0 = CD + 0 8. Additive Inverse Property - by definition,

subtraction means to “add the opposite”.

9. AB = CD 9. Property of Zero for Addition

10. AB > CD 10. Definition of Congruent Segments

7. a) AC > BD b) WX > YZ c) MO > NP

8. a) Statement Reason

1. AB > BD 1. Given

2. BC > DE 2. Given

3. BD > CE 3. Exercise 6 - Part A - Common Segment

4. AB = BD, BD = CE 4. Definition of Congruent Segments

5. AB = CE 5. Transitive Property of Equality

6. AB > CE 6. Definition of Congruent Segments

9. No, Theorem 1 states “If a point lies outside a line, then exactly one plane contains the line and the point.” Point A and collinear

points B, C, D, and E will be on exactly one plane by Theorem1. Point F and collinear points B, C ,D, and E will be on exactly one plane

by Theorem1. However, point A and point F are not necessarily on the same plane, so the planes ABCDE and FBCDE could form a

“vee” or dihedral angle with collinear points B, C, D, and E on the “edge” of the angle.

10. No, The proof only deals with the collinear points B, C, D, and E and the segment AB. The segments CF and FE have no

connection to the proof.


Unit III — Fundamental TheoremsPart B — Theorems About Points and Lines

p. 231 – Lesson 2 — Theorem2: If three different points are on a line, then at most one is between the other two

1. a) If three different points are on a line, then at most one is between the other two.

b)

c) Given: points A, B, and C on line ,d) Prove: B alone lies between A and C

e) Statement Reason

1. Suppose point B is between points A and C, 1. Indirect Proof Assumption.

and point C is between points A and B.


3. AC + CB = AB 3. Postulate 6 - Ruler - Fourth Assumption

4. (AC + BC) + BC = AC 4. Substitution (3 into 2)

5. AC + (BC + BC) = AC 5. Associative Property of Addition

6. AC + (1 + 1) BC = AC 6. Distributive Property of Multiplication to Addition -

7. AC + 2BC = AC 7. Substitution

8. AC + 2BC + -AC = AC + -AC 8. Addition Property for Equations

9. AC + -AC + 2BC = AC + -AC 9. Commutative Property of Addition

10. 0 + 2BC = 0 10. Additive Inverse Property

11. 2BC = 0 11. Identity Property of Addition

12. 12. Multiplication Property of Equality


14. 14. Multiplication Property of 1

15. BC = 0 15. Multiplication Property of Zero (or Substitution)

16. B and C must be the same point 16. Since BC = 0

17. Points A, B, and C are on line , 17. Given

18. This contradiction means our 18. B cannot be between A and C at the same time

assumption is false. C is between A and B.

19. B alone lies between A and C 19. R.A.A.

2. a) line b) line c) plane d) plane e) plane

3. a) always b) never c) always d) sometimes e) never

l

1

2

1

22 BC = 0!

11

20BC =

BC =1

20

A

B

C


• • •

• •

•

4. 5. 6.

7.

8. Statement Reason

1. EX > WG 1. Given

2. EX = WG 2. Definition of Congruent Segments

3. EW + WX = EX 3. Postulate 6 - Ruler - Fourth Assumption

WX + XG = WG

4. EW + WX = WX +XG 4. Substitution

5. EW + WX + -WX = WX + XG + -WX 5. Addition Property of Equality

6. EW + WX + -WX = WX + -WX + XG 6. Commutative Property of Addition

7. EW + 0 = 0 + XG 7. Additive Inverse Property

8. EW = XG 8. Identity Property of Addition

9. GX > HX 9. Given

10. GX = HX 10. Definition of Congruent Segments

11. EW = HX 11. Transitive Property of Equality

(step 8 and 10 - EW = XG, GX = HX, so EW = HX)

12. EW > FW 12. Given

13. EW = FW 13. Definition of Congruent Segments

14. FW = EW 14. Symmetric Property of Equality

15. FW = HX 15. Transitive Property of Equality (Step 14 to 11)

16. FW > HX 16. Definition of Congruent Segments

BD - BC = CD

12 - 4 = CD

8 = CCD

AB CD

AB CD

AB

≅=

8=

W + Y = WY

+ 2 + 1

X X

X 22 = 3 + 6

+ 14 = 3 + 6

X

X X

X + - + 14 + -6 = 3X + - + 6 + -6

X X

0 + 8 = 2 + 0X

8 = 21

2

1

2X

4 =

WY Z

X

X≅WY Z

+ 2 + 12 =

= X

X Z

+ 14 = Z

X

X X

18 = Z

Y + YZ

X

X == Z

12 + YZ = 18

12 + Y

X

ZZ + -12 = 18 + -12

0 + YZ == 6

YZ = 6

FG + GH = FH

24 + 6 = FH

30 = FH

EG FH

EG = F

≅HH

EG = 30

QR + RS = QS

8 + 2 = 4 - 2

8 + 2 + 2 + -2 = 4 - 2 +

x x

x x x 22 + -2

+ 0 = 2 + 0

x

x10

10 2

1

2

1

2= x

5 X

PQ + QR = PR

=

PQ RS

≅PQ RS

PQ

== 2

PQ 2 5 = 10

x

=10 + 8 PR

=18 PR=


• •

•

••

9. Statement Reason

1. GI > HJ 1. Given

2. GI = HJ 2. Definition of Congruent Segments

3. GH + HI = GI 3. Postulate 6 - Ruler - Fourth Assumption

HI + IJ = HJ

4. GH + HI = HI + IJ 4. Substitution Property

5. 5. Addition Property of Equality

6. 6. Commutative Property of Addition

7. GH + 0 = IJ + 0 7. Additive Inverse Property

8. GH = IJ 8. Identity Property of Equality

9. IK > JL 9. Given

10. IK = JL 10. Definition of Congruent Segments

11. IJ + JK = IK 11. Postulate 6 - Ruler - Fourth Assumption

JK + KL = JL

12. IJ + JK = JK + KL 12. Substitution

13. IJ + JK + -JK = JK + KL + JK 13. Addition Property of Equality

14. IJ + JK + -JK = JK + -JK + KL 14. Commutative Property of Addition

15. IJ + 0 = 0 + KL 15. Additive Inverse Property

16. IJ = KL 16. Identity Property of Addition

17. GH = KL 17. Transitive Property of Equality (Statement 8 - Statement 16)

18. GH > KL 18. Definition of Congruent Segments

10. Statement Reason

1. /LMQ > /NMP 1. Given

2. m/LMQ = m/NMP 2. Definition of Congruent Angles

3. m/LMP + m/PMQ = m/LMQ 3. Postulate 7 - Protractor - Fourth Assumption

4. m/NMQ + m/QMP = m/NMP 4. Postulate 7 - Protractor - Fourth Assumption

5. m/LMP + m/PMQ = m/NMQ + m/QMP 5. Substitution

6. m/LMP + m/PMQ + -m/PMQ = 6. Addition Property of Equalitym/NMQ + m/QMP + -m/QMP

7. m/LMP + 0 = m/NMQ + 0 7. Additive Inverse Property

8. m/LMP = m/NMQ 8. Identity Property of Addition

9. /LMP > /NMQ 9. Definition of Angle Congruence.

GH + HI + -HI HI + IJ + -HI=

GH + HI + -HI IJ + HI + -HI=


Unit III — Fundamental TheoremsPart C — Theorems About Segments and Rays

p. 234 – Lesson 1 — Theorem 3: If you have a given ray, then there is exactly one point, at a given distance from the endpoint of the ray.

1. a) If you have a given ray, then there is exactly one point, at a given distance from the end point of the ray.

b)

c) Given: AB and distance d.

d) Prove: Only X is at a distance d from point A on AB

e) Statement Reason

1. AB 1. Given.

2. Zero corresponds to point A. 2. Postulate 6 - Ruler Postulate - First Assumption

3. All other points on the ray correspond 3. Postulate 6 - Ruler Postulate - First Assumption

to positive real numbers. X and B correspond

to positive real numbers.

4. We have “d ” 4. Given

5. | O - X | = d 5. Postulate 6 - Ruler Postulate - Third Assumption

6. Only one point corresponds to d 6. Postulate 6 - Ruler Postulate - Second Assumption

(distance is unique)

2. One

3. Point Q

4. No: The rays do not have the same initial (or end) point, and could go in opposite directions.

5. No.

Case 1: If point T is between point Q and point S, or point S is between point Q and point T, then the rays are the same.

Case 2: Q, T, and A, may not be collinear.

6. No; If point T is between point Q and Point S, or point S is between point Q and point T, then the rays are the same.

11Part C – Theorems About Segments and Rays

A X B

O X b

Q T S Q S T

Q T S Q S T

T

S

Q

7. a)

b)

c)

d)

8. a) yes b) no

9.

BD AB + BD = AD DC AB + BC = z + x = ACBD = AD - AB AB + BD = yBD = y - z AC - (AB + BD) = DC

(z + x) -y = DCz + x -y = DC

10. a)

b)

c) – 9 — (– 3) = – 9 + 3 = – 6 = 6

11. 1. Given


5. Substitution (4 into 3)

8. Addition Property for Inequalities

9. Additive Inverse

10. Identity for Addition

11. Given

15. Addition for Inequality

16. Additive Inverse


21. Distributive Inequality

23. Multiplication Property for Inequalities

24. Property of Zero for Multiplication

25. Multiplication Inverse Property

26. Identity Multiplication

27. a < b means b > a


Q A B

X A B

R S T

A B D

A B D

A B D

– 12 – 6 0

– 12 – 6

– 12 or 0

– 3 or – 90

C

y

z x

P Q

T

midpointmidpoint

28. Multiplication for Inequality

29. Distributive • to +

30. Multiplication for -1

31. Addition for Inequality


33. Identity Addition

35. Substitution



38. Multiplication Property of Zero

39. Multiplication for -1


1. X is between R and T; the coordinates of R 1. Given

and T are zero and t, respectively.

2. X has a coordinate, call it x 2. Postulate 6 - Ruler - First Assumption

3. RX + XT = RT 3. Postulate 6 - Ruler - Fourth Assumption

4. RX = 0 — x = — x = x 4. Postulate 6 - Ruler - Third Assumption

XT = x — t

RT = 0 — t = – t = t

5. x + x — t = t 5. Substitution (4 into 3)

6. x — t < t 6. Definition of >

7. x — t < t and x — t > — t 7. Algebra - If x < k, then x < k and x > – k.

8. t = t 8. Definition of Absolute Value

9. x — t < t and x — t > – t 9. Substitution

Consider x — t > — t

10. 10. Multiplication for Inequality

11. 11. Definition of Subtraction

12. 12. Distributive x to +

13. 13. Multiplication for – 1

14. 14. Addition for Inequality

15. -x + 0 < 0 15. Additive Inverse Property

16. -x < 0 16. Identity for Addition

17. 17. Multiplication for Inequality

18. 18. Property of -1 for Multiplication

19. x > 0 19. Property of Zero for Multiplication

20. The coordinate of X is positive. 20. Definition of Positive (Greater than Zero)


- - < - (-1 1( ) )x t t+

- < - (-1 1( ) )x t t−

- (- - < - (-1 1 1! x t t+ ) ( ) )

- <x t t+

- - < -x t t t t+ + +

(- (- (- 0 1 1) ) )x >

x (- 0 > 1) !

•

•

••

•

•

12. (continued)

Consider x - t < t

21. x — t + t < t + t 21. Addition for Inequality

22. x + 0 < t + t 22. Additive Inverse Property

23. x < t + t 23. Identity for Addition

24. x < (1 + 1) t 24. Distributive x to 0

25. x < 2t 25. Substitution

26. t > 0 26. Given

27. t is a positive number 27. A number greater than zero is positive

28. x < a positive number 28. Substitution (x < 2 • positive number)

29. x could be zero or x could be negative 29. Zero or a negative number is less than a positive number.

30. x cannot be zero or negative 30. It is given that x is between zero and t. So, we have a

contradiction and cannot use this part of the “and” statement.

Unit III — Fundamental TheoremsPart C — Theorems About Segments and Rays

p. 239 – Lesson 2 — Theorem 4: If you have a given line segment, then that segment has exactly one midpoint.

1. a) Theorem 4: If you have a given line segment, then that segment has exactly one midpoint.

b)

c) Given: AB with midpoint C

d) Prove: C is the only midpoint of AB

e) Statement Reason

1. Line segment AB point C is the midpoint of AB 1. Given.

2. AC > CB 2. Definition of Midpoint

3. AC = CB 3. Definition of congruent segments

4. AC + CB = AB 4. Postulate 6 - Ruler Fourth Assumption

5. AC + AC = AB 5. Substitution (3 into 4)

6. 2 AC = AB 6. Collect like terms

7. 7. Multiplication for Equality


9. 9. Identity for Multiplication

10. Suppose D is a different midpoint of AB 10. Indirect Proof Assumption

11. AD > DB 11. Definition of Midpoint

12. AD = DB 12. Definition of Congruent Segments

A C B

1

2

1

2

=

2 AC AB!

11

2AC AB! =

AC AB=

1

2


• •

••

•


1. (continued)

13. AD + DB = AB 13. Postulate 6 - Ruler Fourth Assumption

14. AD + AD = AB 14. Substitution

15. 15. Distributive Property

16. 16. Collect Like Terms

17. 17. Multiplication for Equality

18. 18. Multiplicative Inverse

19. 19. Multiplicative Identity

20. AD = AC 20. Substitution (9 into 19)

21. C and D are the same point 21. Postulate 6 - Ruler - First Assumption

22. The segment has exactly one midpoint. 22. R.A.A.

2. Point Q must be between point M and point N so that m< q < n or n < q < m and MQ > QN (II-B-5)

3. AB = CD

4. Point N

5. Z is the midpoint of XY.

6. Midpoint of AM

7. CD ? EF

8. yes

9. Statement Reason

1. AB 1. Given

2. AB = AB 2. Reflexive Property of Equality

3. AB > AB 3. Definition of Congruent Line Segments

Reflexive property of congruence of line segments


1. AB > CD 1. Given

2. AB = CD 2. Definition of Congruent Segments

3. CD = AB 3. Symmetric for Equality

4. CD > AB 4. Definition of Congruent Line Segments

Symmetric Property of Congruent Line Segments

(1 1) AD = AB+

2 AD = AB!

1

22

1

2AD = AB!

11

2AD = AB!

AD = AB1

2

•

•

• •

••

•


1. AB > CD 1. Given

CD > EF

2. AB = CD 2. Definition of Congruent Line Segments

CD = EF

3. AB = EF 3. Transitivity for Equality

4. AB > EF 4. Definition of Congruent Line Segments

Transitive Property of Congruent Line Segments

12. a)

b)

c) d)

13. 1. Given


3. Property of Zero for Multiplication

4. Substitution

5. Definition of Distance or Postulate 6

6. Transitivity for Equality

7. Definition of Congruent Line Segments

8. Addition for Equality (5)

9. Distributive x to +

10. Substitution or Addition Fact

11. Identity for Multiplication

12. Substitution


14. Definition of Midpoint (7 and 13)

C M D

A

C

D

A

M

B

(answer can vary)

(answer can vary)

2 4 6C M D

12 15 18



14. 1. Given

2. Postulate 6 - Ruler - First Assumption

3. Given

4. Subtraction for Inequality

5. Substitution

6. Definition of Midpoint

7. Definition of Congruent Line Segments

8. Definition of Distance or Postulate 6

9. Substitution (8 into 7)

10. Algebra: If x = k, then x = k or x = – k.

11. Addition for Equality

12. Distributivity x to +

13. Substitution of Equals

14. Commutativity for Addition



17. Multiplication for Equality

18. Multiplicative Inverse

19. Identity Property of One

20. Substitution

21. Substitution

Unit III — Fundamental TheoremsPart D — Theorems About Two Lines

p. 244 – Lesson 1 — Theorem5: If two lines intersect, then exactly one plane contains both lines.

1. a) If two lines intersect, then exactly one plane contains both lines.

b)

c) Given: , and m intersect at T

d) Prove: Exactly one plane contains , and m.

e) Statement Reason

1. , and m intersect at T 1. Given.

2. Point T lies on , and m 2. Definition of Intersecting Lines.

3. Point X lies on ,. 3. Every line contains at least two points.

Point Y lies on m.

4. Points X, Y, and T do not lie on the same line. 4. Step 4 Statement

5. Exactly one plane contains T, X and Y. 5. Exactly one plane contains three different non-collinear points.

6. , is in plane P. 6. Postulate 3 - for any two different points in a plane,

m is in plane P. the line contain them is in the plane.

7. Exactly one plane contains two intersecting lines. 7. Q.E.D.

2. a) False k) True

b) True l) True

c) True m) False

d) True n) True

e) False o) False

f) False p) True

g) True q) False

h) True r) True

i) False s) False

j) False t) True

3. Given: Line , and m intersect at point X.

/UXV > /VXW

Prove: Line , line m.

d) Statement Reason

1. , and m intersect at X 1. Given.

2. /UXW is a straight angle equal to 180º. 2. Definition of straight angle - an angle whose sides are

opposite rays, giving a measure of 180º.


T

m

⊥

19Part D – Theorems About Two Lines

3. (continued)

Statement Reason

3. m/UXV + m/VXW = /UXW 3. Postulate 7 - Protractor - Fourth Assumption.

4. m/UXV + m/VXW = 180º 4. Substitution of Equals (2 into 3)

5. /UXV > /VXW 5. Given

6. m/UXV = m/VXW 6. Definition of Congruent angles.

7. m/UXV + m/UXV = 180 7. Substitution of Equals (6 into 3)

8. (1 + 1) • m/UXV = 180 8. Distributive Property of Multiplication Over Addition.

9. 2m/UXV = 180 9. Substitution of Equals (2 = 1 + 1)

10. 10. Multiplication Property for Equations

11. 11. Substitution of Equals

12. 1 • m/UXV = 90 12. Multiplicative Inverse Property

13. m/UXV = 90 13. Multiplicative Identity

14. /UXV is a right angle 14. Definition of right angle

15. , m 15. Definition of perpendicular lines - lines which intersect

to form a right angle

4. Given: Points R, S, T, and U are collinear

RT > SU

Prove: RS > TU

Statement Reason

1. Points R, S, T, and U are collinear 1. Given

2. RT > SU 2. Given

3. RS + ST = RT 3. Postulate 6 - Ruler - Fourth Assumption.

ST + TU = SU

4. RS + ST > ST + TU 4. Substitution of Equals (3 into 2)

5. RS + ST + -ST > ST + TU + -ST 5. Property of Addition for Inequalities

6. RS + ST + -ST > ST + -ST + TU 6. Commutative Property of Addition

7. RS + 0 > 0 + TU 7. Property of Additive Inverse (a + -a = 0)

8. RS > TU 8. Identity Property of Addition (a + o = a)

5. Given: /YXZ and /YZX are complementary

/XYW > /YZX

/WYZ > /YXZ

Prove: XY YZ

Statement Reason

1. /YXZ and /YZX are complementary 1. Given

1

2180

180

2= = 90!

1

22

1

2UXV = 180! m∠

1

22 9 UXV = 0! m∠

• •

••

5. (continued)

Statement Reason

2. m/YXZ + m/YZX = 90 2. Definition of Complementary Angles

3. /WYZ > /YXZ 3. Given

/XYW > /YZX

4. m/WYZ = m/YXZ 4. Definition of Congruent Angles

m/XYW = m/YZX

5. m/WYZ + m/XYW = 90 5. Substitution of Equals (4 into 2)

6. m/XYW + m/WYZ = 90 6. Commutative Property of Addition

7. m/XYW + m/WYZ = m/XYZ 7. Postulate 7 - Protractor - Fourth Assumption

8. m/XYZ = 90 8. Substitution of Equals (7 into 6)

9. /XYZ is a right angle 9. Definition of Right Angle

10. XY YZ 10. Definition of Perpendicular Lines -

Lines which intersect to form a right angle.

6. Given: m/EBC = m/ECB

Prove: m/ABE = m/ECD

Statement Reason

1. /ABC is a straight angle with a measure of 180º 1. Definition of Straight Angle - an angle whose sides are

/DCB is a straight angle with a measure of 180º opposite rays, giving a measure of 180º

2. m/ABE + m/EBC = m/ABC 2. Postulate 7 - Protractor - Fourth Assumption

m/BCE + m/ECD = m/DCB

3. m/ABE + m/EBC = 180 3. Substitution of Equals (1 into 2)

m/BCE + m/ECD = 180

4. m/ABE + m/EBC = 4. Substitution of Equals (3 into 3)

m/BCE + m/ECD

5. m/EBC = m/ECB 5. Given

6. m/ABE + m/EBC + -m/EBC = 6. Property of Addition for Equality

m/BCE + m/ECD + -m/ECB

7. m/ABE + m/EBC + -m/EBC = 7. Commutative Property of Addition

m/ECD + m/BCE + -m/ECB

8. m/ABE + 0 = m/ECD + 0 8. Property of Additive Inverse (a + -a = o)

9. m/ABE = m/ECD 9. Identity Property of Addition

7. Given: m/DFG = m/HBC = 180

Prove: m/HFG m/HBC

Statement Reason

1. /DFH is a straight angle with a measure of 180º 1. Definition of Straight Angle - an angle whose sides are

opposite rays, giving a measure of 180º

2. m/DFG + m/HFG = m/DFH 2. Postulate 7 - Protractor - Fourth Assumption

3. m/DFG + m/HFG = 180 3. Substitution of Equals (1 into 2)

4. m/DFG + m/HBC = 180 4. Given


AB'


7. (continued)Statement Reason

5. m/DFG + m/HFG = m/DFG + m/HBC 5. Substitution of Equals (4 into 3)

6. m/DFG + m/HFG + -m/DFG = 6. Property of Addition for Equality

m/DFG + m/HBC + -m/DFG

7. m/HBC + m/DFG + -m/DFG = 7. Commutative Property of Addition

m/HBC + m/DFG + -m/DFG

8. m/HFG + 0 = m/HBC + 0 8. Property of Additive Inverse

9. m/HFG = m/HBC 9. Identity Property for Addition (a + o = a)

10. /HFG > /HBC 10. Definition of Angle Congruence

8. Given: AX and BY intersect at point Z as shown.

AX > BY

ZX > ZY

Prove: AZ > BZ

Statement Reason

1. AX and BY intersect at point Z as shown 1. Given.AX > BY

2. AX = BY 2. Definition of Congruent Segments

3. AZ + ZX = AX 3. Postulate 6 - Ruler - Fourth Assumption. BZ + ZY = BY

4. AZ + ZX = BZ +ZY 4. Substitution of Equals (3 into 2)

5. ZX > ZY 5. Given

6. ZX = ZY 6. Definition of Congruent Segments

7. -ZX = -ZY 7. Additive Inverse Property - for every real number a, there exists a unique real number -a such that...

8. AZ + ZX + -ZX = BZ +ZY + -ZY 8. Property of Addition for Equality (4 + 7)

9. AZ + 0 = BZ + 0 9. Additive Inverse Property (a + -a = 0)

10. AZ = BZ 10. Identity Property of Addition

11. AZ > BZ 11. Definition of Congruent Segments

9. Given: PQ and RS intersect at point T.

PQ RS

Prove: /PTS > /STQ

/PTS > /PTR

/RTQ > /QTS

/RTQ > /RTP

Statement Reason

1. PQ and RS intersect at point T. 1. Given.

PQ RS

2. /PTS is a right angle. 2. Definition of Perpendicular lines - lines

/STQ is a right angle. which intersect to form a right angle.

/RTQ is a right angle.

/RTP is a right angle.

9. (continued)

Statement Reason

3. m/PTS = 90 3. Definition of a Right Angle

m/STQ = 90

m/RTQ = 90

m/RTP = 90

4. m/PTS = m/STQ 4. Substitution of Equals (3 into 3)

m/PTS = m/PTR

m/RTQ = m/QTS

m/RTQ = m/RTP

5. /PTS > /STQ 5. Definition of Congruent Angles

/PTS > /PTR

/RTQ > /QTS

/RTQ > /RTP

10. Given: m/ANC = m/ACF

Prove: m/NCD = m/FCD

Statement Reason

1. m/ANC = m/ACF 1. Given.

2. m/ACF + m/FCD = m/ACD 2. Postulate 7 - Protractor - Fourth Assumption -

m/ACN + m/NCD = m/ACD “Angle Addition” Assumption

3. m/ACF + m/FCD = m/ACN + m/NCD 3. Substitution of Equals (3 into 3)

4. m/ACF = m/ACN 4. Symmetric Property for Equality

5. -m/ACF = -m/ACN 5. Additive Inverse Property - for every real number a, there

exists a unique real number -a such that...

6. m/ACF + m/FCD + -m/ACF = 6. Property of Addition for Equality (3 + 5)

m/ACN + m/NCD + -m/ACN

7. m/FCD + m/ACF + -m/ACF = 7. Commutative Property of Addition

m/NCD + m/ACN + -m/ACN

8. m/FCD + 0 = m/NCD + 0 8. Additive Inverse Property (a + -a = o)

9. m/FCD = m/NCD 9. Identity Property of Addition (a + o = a)

10. m/NCD = m/FCD 10. Symmetric Property for Equality

Unit III — Fundamental TheoremsPart D — Theorems About Two Lines

p. 247 – Lesson 2 — Theorem6: If, in a plane, there is a point on a line, then there is exactly one perpendicular to the line,through that point.

1. a) Theorem6: If, in a plane, there is a point on a line, then there is exactly one perpendicular to the line, through that point.

b)M

P

B

A



1.(continued)

c) Given: Point P is on AB in Plane M

d) Prove: Exactly one line in plane M is perpendicular to AB through point P.

e) Statement Reason

1. Point P is on AB 1. Given.

2. There exists a line in plane M, through point P, 2. Postulate 7 - Protractor - Second Assumption

which forms a 90º angle with AB

3. Line , is perpendicular to AB through point P. 3. Definition of Perpendicular

4. Suppose a second line, ,, also forms a 90º 4. Indirect Proof Assumption

angle through point P.

5. Line , is perpendicular to AB through point P. 5. Definition of Perpendicular

6. Line , must be the same as line , 6. Postulate 7 - Protractor - First Assumption - one to one

correspondence with real numbers.

7. Exactly one line in plane M is perpendicular to 7. R.A.A.

AB through point P

2. a) MC MB or MC MA or MC AB or MC BA

b) MC ; AB

c) Line t and CM are the same line. Theorem6 states that for a point on a line there can be only one perpendicular to the line at

that point. Since the two lines are in the same plane, they would have to be the same line.

d) MR and CM are the same line. (Theorem6) All points of MR correspond to all points of CM. Point R is on CM.

e) ; Theorem5 states that if two different lines intersect, then exactly one plane contains both lines. Line p contains point M,

the intersection of AB and CM. Line p must then be perpendicular to both lines (Theorem6)

3. a) False; choose point B not on CD

b) True

c) False; Consider the segment in space. The segment could have an infinite number of perpendicular bisectors.

d) False;

There are an infinite number of lines in plane K perpendicular to AB at midpoint M of AB.

A

M

K

B

1

2

3

4

4. Given: BA AD

BC CD

/2 > /4

Prove: /1 > /3

Statement Reason

1. BA AD 1. Given

BC CD

2. /BAD is a right angle 2. Definition of Perpendicular Lines (line segments)

/BCD is a right angle

3. m/BAD = 90º 3. Definition of Right Angle

m/BCD = 90º

4. m/1 + m/2 = m/BAD 4. Postulate 7 - Protractor - Fourth Assumption.

m/3 + m/4 = m/BCD

5. m/1 + m/2 = 90 5. Substitution of Equals (3 into 4)

m/3 + m/4 = 90

6. m/1 + m/2 = m/3 + m/4 6. Substitution of Equals (4 into 4)

7. /2 > /4 7. Given

8. m/2 = m/4 8. Definition of Angle Congruence

9. -m/2 = -m/4 9. Property of Additive Inverse - For every real number a,

there exists a unique real number -a, such that...

10. m/1 + m/2 + -m/2 = 10. Addition Property for Equality (6 + 9)

m/3 + m/4 + -m/4

11. m/1 + 0 = m/3 + 0 11. Additive Inverse Property (a + -a = o)

12. m/1 = m/3 12. Identity Property for Addition (a + o = a)

13. /1 > /3 13. Definition of Angle Congruence

5. Given: /ABD and /DBC are complementary;

/ADB > /ABC

Prove: BD AC

Statement Reason

1. /ABD and /DBC are complementary 1. Given

2. m/ABD + m/DBC = 90 2. Definition of Complementary Angles

3. m/ABD + m/DBC = m/ABC 3. Postulate 7 - Protractor - Fourth Assumption.

4. m/ABC = 90 4. Substitution of Equals (3 into 2)

5. /ADB > /ABC 5. Given

6. m/ADB = m/ABC 6. Definition of Congruent Angles

7. m/ADB = 90 7. Substitution of Equals

8. /ADB is a right angle 8. Definition of Right Angle

9. BD AC 9. Definition of Perpendicular Lines (line segments)



6. Given: /CBG > /EFA

/CBG and /EFA are supplementary

Prove: CD AG

Statement Reason

1. /CBG and /EFA are supplementary 1. Given

2. m/CBG + m/EFA = 180 2. Definition of Supplementary Angles

3. /CBG > /EFA 3. Given

4. m/CBG = m/EFA 4. Definition of Congruent Angles

5. m/CBG + m/CBG = 180 5. Substitution of Equals

6. (1 + 1) • m/CBG = 180 6. Distributive Property of Multiplication Over Addition

7. 2m/CBG = 180 7. Substitution of Equals (6 into 7; 1 + 1 =2)

8. 8. Property of Multiplication for Equality

9. 9. Property of Multiplicative Inverse

10. 10. Identity Property for Multiplication - (1 • a = a)

11. m/CBG = 90 11. Substitution of Equals

12. /CBG is a right angle 12. Definition of Right Angle

13. CD AG 13. Definition of Perpendicular Lines

7. a) m/MNP = m/ONP since both are right angles formed by two perpendicular lines.

b) MO NP since /MNP and /ONP are two equal angles forming a straight angle. Each will equal 90º. Each will be a right angle.

8.135;

The sum of the two obtuse angles must equal 270º if the non common sides form a right angle. (360 - 90 = 270). Two angles whose

sum is 270, cut in half by angle bisectors, would be 135.

0 into 11 ; 90 = 1801

2

1

22

1

2CBG = 180! m∠

11

2CBG = 180! m∠

m∠CBG = 1801

2

C

OA

B

non-common sides

•

• •

••

•

1

9.yes;

10. a) yes

b) no

Unit III — Fundamental TheoremsPart E — Theorems About Angles – Part 1 (One Angle)

p. 250 – Lesson 1 — Theorem7: If, in a half-plane, there is a ray in the edge of the half-plane, then there is exactly one other raythrough the endpoint of the given ray, such that the angle formed by the two rays has a givenmeasure.

1. a) Theorem7: If, in a half-plane, there is a ray in the edge of the half-plane, then there is exactly one other ray through the

endpoint of the given ray, such that the angle formed by the two rays has a given measure.

b)

c) Given: AB in the edge of half-plane H.

d) Prove: There is only one ray, which forms an angle with AB such that the measure of the angle is r.

P

M

N

45

C

AB

H

r

B

A

A

C

H

P

Y

x

z


1. (continued)

e) Statement Reason

1. AB is a ray in the edge of a half-plane. 1. Given

2. The real number r is the desired angle measure. 2. Given

3. Let AB correspond to the number zero. 3. Postulate 7 – Protractor – First Assumption

4. There is only one ray, call it AC, which corresponds 4. Postulate 7 – Protractor – First Assumptionto the real number r .

5. m/BAC = r - 0 or 0 - r 5. Postulate 7 – Protractor – Third Assumption

6. r - 0 = r = r 6. Definition of Absolute Value

7. There is only one ray, such that the angle formed 7. Q. E. D.with the given ray, has a given measure.

2. Given: /ABC

Prove: /ABC > /ABC

Statement Reason

1. /ABC 1. Given

2. m/ABC = m/ABC 2. Reflexive Property of Equality

3. /ABC > /ABC 3. Definition of Congruent Angles

The reflexive property for congruence of angles describes this relationship.

3. Given: /ABC > /DEF

Prove: /DEF > /ABC

a) If /ABC > /DEF, then /DEF > /ABC

b) Statement Reason

1. /ABC > /DEF 1. Given

2. m/ABC = m/DEF 2. Definition of Congruent Angles

3. m/DEF = m/ABC 3. Symmetric Property of Equality

4. /DEF > /ABC 4. Definition of Congruent Angles

The symmetric property for congruence of angles describes this relationship.

4. Given: /ABC > /DEF/DEF > /GHI

Prove: /ABC > /GHI

a) If /ABC > /DEF and /DEF > /GHI, then /ABC > /GHI

b) Statement Reason

1. /ABC > /DEF 1. Given/DEF > /GHI

2. m/ABC = m/DEF 2. Definition of Congruent Anglesm/DEF = m/GHI

3. m/ABC = m/GHI 3. Transitive Property of Equality

4. /ABC > /GHI 4. Definition of Congruent Angles

The transitive property for congruence of angles describes this relationship.

27Part E – Theorems About Angles – Part 1 (One Angle)

5. m/APC 6. m/DPA 7. m/BPC 8. m/CPD 9. 87 + 29 = 116 10. 78 – 49 = 29

11.

c) m/ABC = 135°

d) Supplementary

e) Opposite

f) 180°

12. The sum must be 180°

13. Given: BC CD; /FBR > /DCF

/CBF > /FCB

Prove: m/CBR = 90

Statement Reason

1. BC CD 1. Given

2. /DCB is a right angle 2. Definition of Perpendicular Lines (Segment to Ray)

3. m/DCB = 90° 3. Definition of Right Angle

4. m/DCF + m/FCB = m/DCB 4. Postulate 7 – Protractor – Fourth Assumption

5. m/DCF + m/FCB = 90 5. Substitution (3 into 4)

6. /FBR > /DCF 6. Given/CBF > /FCB

7. m/FBR = m/DCF 7. Definition of Congruent Anglesm/CBF = m/FCB

8. m/FBR + m/CBF = 90 8. Substitution (7 into 5)

9. m/CBF + m/FBR = 90 9. Commutative Property of Addition

10. m/CBF + m/FBR = m/CBR 10. Postulate 7– Protractor – Fourth Assumption

11. 90 = m/CBR 11. Substitution (9 into 10)

12. m/CBR = 90 12. Symmetric Property of Equality

14. a) 140; Exercise 12 – non-common sides opposite rays with /DPB.

b) 40; Exercise 12 – non-common sides opposite rays with /APD.

c) Yes; Definition of congruent angles

d) No

e) Yes


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Unit III — Fundamental TheoremsPart E — Theorems About Angles – Part 1 (One Angle)

p. 253 – Lesson 2 — Theorem8: If in a half-plane, you have an angle, then that angle has exactly one bisector.

1. a) Theorem 8: If in a half-plane, you have an angle, then that angle has exactly one bisector.

b)

c) Given: OB is a bisector of /COA in half-plane J

d) Prove: OB is the only bisector of /AOC

e) Statement Reason

1. OB is a bisector of /AOC 1. Given

2. /AOB > /BOC 2. Definition of Angle Bisector

3. m/AOB = m/BOC 3. Definition of Congruent Angles

4. OB lies between OA and OC 4. Definition of Angle Bisector

5. m/AOB + m/BOC = m/AOC 5. Postulate 7 – Protractor – Assumption Four –Angle Addition Assumption

6. m/AOB + m/AOB = m/AOC 6. Substitution (3 into 5)

7. (1 + 1) • m/AOB = m/AOC 7. Distributive Property of Multiplication Over Addition

8. 2 • m/AOB = m/AOB 8. Substitution

9. 9. Multiplication Property for Equality


11. 11. Identity Property of Multiplication (1 • a = a)

12. Suppose we have another bisector of /AOC, 12. Indirect Proof Assumptioncall it OD

13. /AOD > /DOC 13. Definition of Angle Bisector

14. m/AOD = m/DOC 14. Definition of Congruent Angles

15. OD lies between OA and OC 15. Definition of Angle Bisector

16. m/AOD + m/DOC = m/AOC 16. Postulate 7 – Protractor – Assumption Four –Angle Addition Assumption

17. m/AOD + m/AOD = m/AOC 17. Substitution (14 into 16)

18. (1 + 1) m/AOD = m/AOC 18. Distributive Property of Multiplication Over Addition

19. 2 • m/AOD = m/AOC 19. Substitution (1 + 1 = 2)



22. 22. Identity Property of Multiplication

23. m/AOB = m/AOD 23. Substitution (11 into 22)

24. OB = OD (same ray) 24. Postulate 7 – Protractor – Second Assumption – Unique Measure

25. OB is the only bisector of m/AOC 25. R. A. A.

29Part E – Theorems About Angles – Part 1 (One Angle)0

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2m AOB = m AOC! ∠ ∠

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2m AOD = m AOC! ∠ ∠

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2m AOD = m AOC! ∠ ∠

m AOD = m AOC∠ ∠1

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11

2m AOB = m AOC! ∠ ∠

m AOB = m AOC∠ ∠1

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2. /QSR; Definition of Angle Bisector

3. a) QSP b) Bisector

4. a) 60° b) 75° c) 70°

5. a)

b) Broken rays are bisectors c) 90°d) m/NMP + m/PML is 180° since together the angles form a straight angle. So,

6. m/RSZ = 18m/NSZ = 54

7. /VSR > /VST/URT > /URS We could deduce that m/VST = m/URT if we knew /RST > /SRT or /RSV > /SRU.

8. Given: EG bisects /DEFSW bisects /RSTm/DEG = m/RSW

Prove: m/DEF = m/RST

Statement Reason

1. EG bisects /DEF 1. GivenSW bisects /RST

2. /DEG > /GEF 2. Definition of Angle Bisector/RSW > /WST

3. m/DEG = m/GEF 3. Definition of Congruent Anglesm/RSW = m/WST

4. m/DEG + m/RSW = m/GEF + m/WST 4. Addition Property for Equality ( 3 + 3)

5. m/DEG = m/RSW 5. Given

6. m/DEG + m/DEG = m/DEG + m/WST 6. Substitution (5 and 3 into 4)

7. m/DEG + m/DEG + -m/DEG = 7. Addition Property for Equalitym/DEG + m/WST + -m/DEG

8. m/DEG + m/DEG + -m/DEG = 8. Commutative Property of Additionm/DEG + -m/DEG + m/WST

9. m/DEG + 0 = 0 + m/WST 9. Property of Additive Inverse (a + -a = 0)

10. m/DEG = m/WST 10. Identity Property of Addition (a + 0 = a)

11. m/DEG + m/GEF = m/DEF 11. Postulate 7– Protractor – Fourth Assumption – Anglem/RSW + m/WST = m/RST Addition Assumption

12. m/DEG + m/DEG = m/DEF 12. Substitution (3, 5, 10 into 11)m/DEG + m/DEG = m/RST

13. m/DEF = m/RST 13. Substitution (12 into 12)


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280 90 m NMP + m PML = 1! ∠ ∠ =

9. Given: BX bisects /ABC

Prove: m/ABX = m/ABC

m/XBC = m/ABC

Statement Reason

1. BX bisects /ABC 1. Given

2. /ABX > /XBC 2. Definition of Angle Bisector

3. m/ABX = m/XBC 3. Definition of Congruent Angles

4. m/ABX + m/XBC = m/ABC 4. Postulate 7 – Protractor – Fourth Assumption – Angle AdditionAssumption

5. m/ABX + m/ABX = m/ABC 5. Substitution (3 into 4)m/XBC + m/XBC = m/ABC

6. (1 + 1) • m/ABX = m/ABC 6. Distributive Property of Multiplication Over Addition(1 + 1) • m/XBC = m/ABC

7. 2 • m/ABX = m/ABC 7. Substitution (1 + 1 = 2)2 • m/XBC = m/ABC



10. 10. Identity Property for Multiplication

10. Given: VP bisects /SVTVQ bisects /RVT

Prove: /RVQ is a complement of /SVP

Statement Reason

1. VP bisects /SVT 1. GivenVQ bisects /RVT

2. /RVQ > /QVT 2. Definition of Angle Bisector/SVP > /PVT

3. m/RVQ = m/QVT 3. Definition of Congruent Anglesm/SVP = m/PVT

4. /RVS is a straight angle with a measure of 180° 4. Definition of Straight Angle – An angle whose sides are oppositerays, giving a measure of 180°

5. m/RVT + m/TVS = m/RVS 5. Postulate 7 – Protractor – Fourth Assumption – Angle AdditionAssumption

6. m/RVQ + m/QVT = m/RVT 6. Postulate 7 – Protractor – Fourth Assumption – Angle Additionm/TVP + m/PVS = m/TVS Assumption


1

2

1

21

2

1

2

2m ABX = m ABC

2m XBC =

! ∠ ∠

∠ m ABC∠

11

2

11

2

m ABX = m ABC

m XBC = m

! ∠ ∠

∠ ∠AABC

m ABX = m ABC

m XBC = m ABC

∠ ∠

∠ ∠

1

21

2

1

21

2

• •

••

• •

••

•

•

7. m/RVQ + m/QVT + m/TVP + m/PVS = 7. Substitution (6 into 5)m/RVS

8. m/RVQ + m/RVQ + m/SVP + m/PVS = 8. Substitution (3 into 7)m/RVS

9. m/RVQ + m/RVQ + m/PVS + m/PVS = 180 9. Substitution

10. (1 + 1) • m/RVQ + (1 + 1) • m/PVS = 180 10. Distributive Property of Multiplication over Addition

11. 2 • m/RVQ + 2 • m/PVS = 180 11. Substitution (1 + 1 = 2)

12. 2(m/RVQ + m/PVS) = 180 12. Distributive Property of Multiplication over Addition




16. m/RVQ + m/PVS = 90 16. Substitution of ( )

17. /RVQ and /PVS are complementary angles 17. Definition of Complementary Angles – Two angles, the sum of

whose measures is 90

11. Given: /APF > /CPDPE bisects /DPF

Prove: PE AC

Statement Reason

1. /APC is a straight angle with a measure of 180° 1. Definition of Straight Angle – An angle whose sides are oppositerays giving a measure of 180°

2. m/APE + m/EPC = m/APC 2. Postulate 7 – Protractor – Fourth Assumption – Angle AdditionAssumption

3. m/APE + m/EPC = 180° 3. Substitution (1 into 2)

4. m/APD + m/DPE = m/APE 4. Postulate 7 – Protractor – Fourth Assumption – Angle Additionm/EPF + m/FPC = m/EPC Assumption

5. m/APD + m/DPE + m/EPF + m/FPC = 180 5. Substitution (4 into 3)

6. /APF > /CPD 6. Given

7. m/APF = m/CPD 7. Definition of Congruent Angles

8. m/APD + m/DPF = m/APF 8. Postulate 7 – Protractor – Fourth Assumption – Angle Additionm/CPF + m/FPD = m/CPD Assumption

9. m/APD + m/DPF = m/CPF + m/FPD 9. Substitution (8 into 7)

10. m/APD + m/DPF + -m/DPF = m/CPF 10. Addition Property for Equality+ m/DPF + -m/DPF

11. m/APD + 0 = m/CPF + 0 11. Additive Inverse Property (a + -a = 0)

*12. m/APD = m/CPF 12. Identity Property for Addition (a + 0 = a)

13. PE bisects /DPF 13. Given

14. /DPE > /EPF 14. Definition of Angle Bisector

*15. m/DPE = m/EPF 15. Definition of Congruent Angles

16. m/APD + m/DPE + m/DPE + m/APD = 180 16. Substitution (15 and 12 into 5)


1

2

1

22 m RVQ + m PVS 180! ∠ ∠( ) =

1 m RVQ + m PVS 180! ∠ ∠( ) = 1

2

m RVQ + m PVS 180∠ ∠ = 1

21

2180 = 90!

• •

•

•

•

•

17. m/APD + m/APD + m/DPE + m/DPE = 180 17. Commutative Property of Addition

18. (1 + 1) m/APD + (1 +1) m/DPE = 180 18. Distributive Property of Multiplication Over Addition

19. 2 • m/APD + 2 • m/DPE = 180 19. Substitution (1 + 1 = 2)

20. 2(m/APD + m/DPE) = 180 20. Distributive Property of Multiplication Over Addition




24. m/APD + m/DPE = 90 24. Substitution ( )

25. m/APE = 90 25. Substitution (4 into 24)

26. /APE is a right angle 26. Definition of Right Angle

27. PE AC 27. Definition of Perpendicular Lines (ray to a line)


1

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1 m APD + m DPE 180! ∠ ∠( ) =1

2

m APD + m DPE 180∠ ∠ =1

21

2180 = 90!

• •

••

•

•

34 Unit III – Fundamental Theorems34 Unit III – Fundamental Theorems

Unit III – Fundamental TheoremsPart F — Theorems About Angles - Part 2 (Two Angles)

p. 256 – Lesson 1 — Theorem 9: If two adjacent acute angles have their exterior sides in perpendicular lines, then the two angles are complementary.

1. a) Theorem 9 – If two adjacent acute angles have their exterior sides in perpendicular lines, then the two angles are complementary.

b)

c) Given: /AOB and /BOC adjacent acute angles with d) Prove: /AOB and /BOC are complementaryexterior sides that are in perpendicular lines, ,and m.

e)

1. , m 1. Given

2. /AOC is a right angle 2. Definition of Perpendicular Lines

3. m/AOC = 90 3. Definition of Right Angle

4. m/AOB + m/BOC = m/AOC 4. Postulate 7 - (Protractor) Angle-Addition Assumption.

5. m/AOB + m/BOC = 90 5. Substitution of Equals (3 into 4)

6. /AOB and /BOC are complementary angles. 6. Definition of Complementary Angles

2. angle xcomplement 90 – xsupplement 180 – x

The angle measures 30O

3. Given: RU RS; TU TS and m/URT = m/UTRProve: m/RTS = m/TRS

1. RU RS; TU TS 1. Given

2. /URT and /TRS are complementary 2. Theorem 9

3. /UTR and /RTS are complementary 3. Theorem 9

4. m/URT+ m/TRS = 90 4. Definition of Complementary Angles

5. m/UTR + m/RTS = 90 5. Definition of Complementary Angles

6. m/URT + m/TRS = m/UTR + m/RTS 6. Substitution of Equals (3 into 3)

7. m/URT = m/UTR 7. Given

8. m/URT + m/TRS – m/URT = m/UTR 8. Subtraction Property for Equality

+ m/RTS – m/UTR

9. m/URT + m/TRS + —m/URT = m/UTR 9. Definition of Subtraction

+ m/RTS + —m/UTR

10. m/URT + —m/URT + m/TRS = m/UTR 10. Commutative Property of Addition

+ —m/UTR + m/RTS

11. 0 + m/TRS = 0 + m/RTS 11. Additive Inverse Property

12. m/TRS = m/RTS 12. Identity Property of Addition

13. m/RTS = m/TRS 13. Symmetric Property of Equality

>

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4. Given: OA OC; OB ODProve: /DOC > /AOB

1. OA OC; OB OD 1. Given

2. /AOB and /BOC are complementary 2. Theorem 9

3. /DOC and /BOC are complementary 3. Theorem 9

4. m/AOB + m/BOC = 90 4. Definition of Complementary Angles

5. m/DOC + m/BOC = 90 5. Definition of Complementary Angles

6. m/AOB + m/BOC = m/DOC + m/BOC 6. Substitution of Equals (5 into 4)

7. m/BOC = m/BOC 7. Reflexive Property of Equality

8. m/AOB + m/BOC – m/BOC = m/DOC 8. Subtraction Property for Equality

+ m/BOC – m/BOC

9. m/AOB + m/BOC + —m/BOC = m/DOC 9. Definition of Subtraction

+ m/BOC + —m/BOC

10. m/AOB + 0 = m/DOC + 0 10. Additive Inverse Property

11. m/AOB = m/DOC 11. Identity Property of Addition

12. m/DOC = m/AOB 12. Symmetric Property of Equality

13. /DOC > /AOB 13. Definition of Angle Congruence

5. Given: ST SR<XYZ is a complement of <RSV

Prove: /XYZ > /VST

1. ST SR 1. Given

2. /RSV and /VST are complementary angles 2. Theorem 9

3. /XYZ is a complement of /RSV 3. Given

4. m/RSV + m/VST = 90 4. Definition of Complementary Angles

5. m/XYZ + m/RSV = 90 5. Definition of Complementary Angles

6. m/RSV + m/VST = m/XYZ + m/RSV 6. Substitution of Equals (3 into 3)

7. m/RSV = m/RSV 7. Reflexive Property of Equality

8. m/RSV + m/VST – m/RSV = m/XYZ 8. Subtraction Property for Equality

+ m/RSV – m/RSV

9. m/RSV + m/VST + —m/RSV = m/XYZ 9. Definition of Subtraction

+ m/RSV + —m/RSV

10. m/VST + —m/RSV + m/RSV = m/XYZ 10. Commutative Property of Addition

+ —m/RSV + m/RSV

11. m/VST + 0 = m/XYZ + 0 11. Additive Inverse Property

12. m/VST = m/XYZ 12. Identity Property of Addition

13. m/XYZ = m/VST 13. Symmetric Property of Equality

14. /XYZ > /VST 14. Definition of Congruent Angles

> >

> >

>

>

35Part F – Theorems About Angles – Part 2 (Two Angles)


6. Given: /2 > /3/1 and /3 are complementary

Prove: nRUT is a right triangle

1. /1 and /3 are complementary 1. Given

2. m/1 + m/3 = 90 2. Definition of Complementary Angles

3. /2 > /3 3. Given

4. m/2 = m/3 4. Definition of Congruent Angles

5. m/1 + m/2 = 90 5. Substitution of Equals

6. m/1 + m/2 = m/RUT 6. Postulate 7 (Protractor) Angle-Addition Assumption

7. 90 = m/RUT 7. Substitution of Equals

8. /RUT is a right angle 8. Definition of Right Angle

9. nRUT is a right triangle 9. Definition of Right Triangle

7. ORm/2 + m/3 = 90so, m/2 = 90 – m/3

m/1 + m/2 = 180so, m/1 + 90 – m/3 = 180m/1 – m/3 = 90 (substitution)

8. a) If two angles are complementary, then the angles are adjacent acute angles with their exterior sides in perpendicular lines.b) yesc) nod) false


p. 258 – Lesson 2 — Theorem 10: If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary.

1. a) Theorem 10 – If the exterior sides of two adjacent angles are opposite rays, then the angles are supplementary.

b)

c) Given: /ADC and /COB are adjacent acute angles with d) Prove: /AOC and /COB are supplementaryexterior sides that are opposite rays.

e)

1. /AOC and /COB are adjacent angles with 1. Givenexterior sides that are opposite rays.

2. OC lies between OA and OB 2. Definition of Adjacent Angles

3. m/AOC + m/COB = m/AOB 3. Postulate 7 (Protractor) Angle-Addition Assumption

4. /AOB is a straight angle 4. Definition of Straight Angle - An Angle whose sides are opposite rays

5. m/AOB = 180 5. Definition of Straight Angle - An Angle whose sides are opposite rays

6. m/AOC + /COB = 180 6. Substitution of Equals

7. /AOC and /COB are supplementary 7. Definition of Supplementary Angles

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2. a) Theorem 10 – Special Case – If one angle of a linear pair is a right angle, then the other angle is a right angle

b)

c) Given: /AOC and /COB are a linear pair d) Prove: /COB is a right angle/AOC is a right angle

e)

1. /AOC and /COB are a linear pair 1. Given

2. /AOC and /COB are adjacent angles with exterior 2. Definition of Linear Pairsides opposite rays

3. /AOC and /COB are supplementary 3. Theorem 10

4. m/AOC + m/COB = 180 4. Definition of Supplementary Angles

5. m/AOC is a right angle 5. Given

6. m/AOC = 90 6. Definition of Right Angle

7. 90 + m/COB = 180 7. Substitution of Equals

8. 90 + m/COB + –90 = 180 + –90 8. Addition Property for Equality

9. m/COB + 90 + –90 = 180 + –90 9. Commutative Property of Addition

10. m/COB + 0 = 180 + –90 10. Additive Inverse Property

11. m/COB = 180 + –90 11. Identity Property of Addition

12. m/COB = 90 12. Substitution of Equals

13. /COB is a right angle 13. Definition of Right Angle

3. a) 180 – 30 = 150b) 180 – 90 = 90c) 180 – 178 = 2

4. The supplement of an acute angle is an obtuse angle.The supplement of an obtuse angle is an acute angle.

5. 180 – n

6. The supplement is 90O more than the complement.

7.

8.

x is the measure of the angley is the measure of the supplementz is the measure of the complement

B

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x + y = 180

x + z = 90

y = 3z

180 - x = 3 90 - x

180 - x = 270 - 3x

2

( )

xx = 90

x = 45


9. Given: Two angles are supplementary and congruent Prove: Each angle is a right angle

Case #1Given: /AOC and /COB are supplementary Prove: /AOC and /COB are right angles

/AOC > /COB

1. /AOC and /COB are supplementary 1. Given

2. m/AOC + m/COB = 180 2. Definition of Supplementary Angles

3. /AOC > /COB 3. Given

4. m/AOC = m/COB 4. Definition of Congruent Angle

5. m/AOC + /AOC = 180 5. Substitution of Equals

6. (1 + 1) m/AOC = 180 6. Distributive Property of Multiplication Over Addition

7. 2 • m/AOC = 180 7. Substitution of Equals

8. 1/2 • 2m/AOC = 1/2 • 180 8. Multiplication Property for Equality

9. 1 • m/AOC = 1/2 • 180 9. Multiplicative Inverse Property

10. m/AOC = 1/2 • 180 10. Identity Property of Multiplication

11. m/AOC = 90 11. Substitution of Equals

12. /AOC is a right angle 12. Definition of Right Angle

13. /AOC and /COB are a linear pair 13. Definition of Linear Pair - (As observed in our case 1 diagram)

14. /COB is a right angle 14. Corollary to Theorem 10 - If one angle of a linear pair is a rightangle, then the other angle is a right angle.

Case #2Given: /AOC and /BPD are supplementary Prove: /AOC and /BPD are right angles

/AOC > /BPD

1. /AOC and /BPD are supplementary 1. Given

2. m/AOC + m/BPD = 180 2. Definition of Supplementary Angles

3. /AOC > /BPD 3. Given

4. m/AOC = m/BPD 4. Definition of Congruent Angle

5. m/AOC + /AOC = 180 or m/BPD + /BPD = 180 5. Substitution of Equals

6. (1 + 1) m/AOC = 180 or (1 + 1) m/BPD = 180 6. Distributive Property of Multiplication Over Addition

7. 2 • m/AOC = 180 or 2 • m/BPD = 180 7. Substitution of Equals

8. 1/2 • 2m/AOC = 1/2 • 180 or 1/2 • 2m/BPD = 1/2 • 180 8. Multiplication Property for Equality

9. 1 • m/AOC = 1/2 • 180 or 1 • m/BPD = 1/2 • 180 9. Multiplicative Inverse Property

10. m/AOC = 1/2 • 180 or m/BPD = 1/2 • 180 10. Identity Property of Multiplication

11. m/AOC = 90 or m/BPD = 90 11. Substitution of Equals

12. /AOC is a right angle or /BPD is a right angle 12. Definition of Right Angle

12

3

BA

C

O BA

C

O

yx

BA

C

O C

A

O

Cm

C

A

B Z

X

Y

DP -1 0 1 2

123

B

C

BA

C

O

B

C

O C

A

O

C

Z

X

Y

0 1 2

123

C

B

A

O

m

?

12

3

BA

C

O A

C

O

y=x+20x yx

BA

C

O

A

O

D

B

P CA

B

O

D

m

C

A

B Z

X

Y

B

A

O

DC

P -1 0 1 2

123


10. Given: Two lines are perpendicular Prove: The angles formed are congruent, adjacent angles

Given: AC BD Prove: /AOB > /BOC; /BOC > /COD/COD > /DOA; /DOA > /AOB

1. AC BD 1. Given

2. /AOB is a right angle 2. Definition of Perpendicular Lines

3. /AOB and /BOC form a linear pair 3. Definition of Linear Pair/BOC and /COD form a linear pair/COD and /DOA form a linear pair/DOA and /AOB form a linear pair

4. /AOB and /BOC are adjacent angles 4. Definition of Linear Pair - ray between exterior sides and exterior /BOC and /COD are adjacent angles sides opposite rays./COD and /DOA are adjacent angles/DOA and /AOB are adjacent angles

5. /BOC, /COD, and /DOA are right angles 5. Corollary to Theorem 10 - If one angle of a linear pair is a right angle, then the other angle is a right angle.

6. m/AOB = 90, m/BOC = 90, m/COD = 90, 6. Definition of Right Anglem/DOA = 90

7. m/AOB = m/BOC = m/COD = m/DOA 7. Substitution

8. /AOB > /BOC > /COD > /DOA 8. Definition of Congruent Angles

11. Given: /1 and /2 are a linear pair/1 > /2

Prove: Line m Line n

1. /1 and /2 are a linear pair 1. Given

2. /1 and /2 are adjacent angles whose exterior 2. Definition of Linear Pairsides are opposite rays

3. /1 and /2 are supplementary 3. Theorem 10

4. m/1+ m/2 = 180 4. Definition of Supplementary Angles

5. /1 > /2 5. Given



8. (1 + 1) m/2 = 180 8. Distributive Property of Multiplication over Addition

9. 2 • m/2 = 180 9. Substitution of Equals

10. 1/2 • 2m/2 = 1/2 • 180 10. Multiplication of Equality

11. 1 • m/2 = 1/2 • 180 11. Multiplicative Inverse

12. m/2 = 1/2 • 180 12. Identity of Multiplication

13. m/2 = 90 13. Multiplication Fact or Substitution

14. /2 is a right angle 14. Definition of Right Angle

15. m n 15. Definition of Perpendicular Lines

>

>

>

>

CO

m

?2

BA O BA O

y=x+20x yx

BA

C

O C

A

O

D

B

P CA

B

O

D

m

C

A

B Z

X

Y

B

A

O

DC

P -1 0 1 2

123


12. If two congruent angles form a linear pair, then the intersecting lines forming the angles are perpendicular.

13. a) If two angles are supplementary, then the two angles are adjacent angles with exterior sides in opposite rays.b) Noc) False


p. 261 – Lesson 3 — Theorem 11: If you have right angles, then those right angles are congruent.

1. a) Theorem 11 – If you have right angles, then those right angles are congruent.

b)

c) Given: /ABC and /XYZ are right angles d) Prove: /ABC > /XYZ

e)

1. /ABC is a right angle 1. Given

2. /XYZ is a right angle 2. Given

3. m/ABC = 90 3. Definition of Right Angle

4. m/XYZ = 90 4. Definition of Right Angle

5. m/ABC = m/XYZ 5. Substitution of Equals

6. /ABC > /XYZ 6. Definition of Congruence

2. /AQC, /CQE, /EQG, /GQA, /BQD, /DQF, /FQH, /HQB

3.

1. m n 1. Given

2. /AEB, /BEC, /CED, and /DEA are right angles 2. Definition of Perpendicular

3. /AEB > /BEC > /CED > /DEA 3. Theorem 11

4.

1. /BOA is a right angle; /BOC is a right angle 1. Given

2. /BOA > /BOC 2. Theorem 11

3. m/BOA = m/BOC 3. Definition of Congruent Angles

4. /BOE > /BOD 4. Given

5. m/BOE = m/BOD 5. Definition of Congruent Angles

6. m/BOE = m/BOA + m/AOE 6. Postulate 7 (Protractor) - Angle-Addition Assumption

m/BOD = m/BOC + m/COD

7. m/BOA + m/AOE = m/BOC + m/COD 7. Substitution of Equals

8. m/BOA + m/AOE –m/BOA = m/BOC 8. Subtraction Property for Equality

+ m/COD – m/BOC

9. m/BOA + m/AOE + –m/BOA = m/BOC 9. Definition of Subtraction (a – a = a + –a)

+ m/COD + – m/BOC

10. m/BOA + –m/BOA + m/AOE = m/BOC 10. Commutative Property of Addition

+ – m/BOC + m/COD

11. 0 + m/AOE = 0 + m/COD 11. Additive Inverse Property (a + –a = 0)

12. m/AOE = m/COD 12. Identity Property for Addition

13. /AOE > /COD 13. Definition of Congruent Angles

>

12

3

BA

C

O BA

C

O

yx

BA

C

O C

A

O

C

B

O

D

m

C

A

B Z

X

Y

DC

P -1 0 1 2

123


5.

1. /MON is a right angle 1. Given

2. /RTS is a right angle 2. Given

3. /MON > /RTS 3. Theorem 11

4. m/MON = m/RTS 4. Definition of Congruent Angles

5. m/MON = m/1 + m/2 5. Postulate 7 (Protractor) Angle-Addition Assumption

6. m/RTS = m/3 + m/4 6. Postulate 7 (Protractor) Angle-Addition Assumption

7. m/1 + m/2 = m/3 + m/4 7. Substitution of Equals

6. Not necessarily; Not necessarily; we need to know that one pair, either /1 and /3 or /2 and /4, is congruent; 45 degrees

7.

1. AB CD; XC CD; YD CD 1. Given

2. /ABC is a right angle; /ABD is a right angle 2. Definition of Perpendicular

/CDY is a right angle; /DCX is a right angle

3. /ABC > /ABD > /CDY > /DCX 3. Theorem 11

4. m/ABC = m/ABD = m/CDY = m/DCX 4. Definition of Congruent Angles

5. m/ABC = m/1 + m/3 5. Postulate 7 (Protractor) Angle-Addition Assumption

m/ABD = m/2 + m/4

6. m/1 + m/3 = m/CDY 6. Substitution Principle

m/2 + m/4 = m/DCX

8.

1. AC AD; AC CB 1. Given

2. /CAD is a right angle; /ACB is a right angle 2. Definition of Perpendicular

3. /CAD > /ACB 3. Theorem 11

4. m/CAD = m/ACB 4. Definition of Congruent Angles

5. /1 > /2 5. Given


7. m/CAD + m/1 = m/ACB + m/2 7. Addition Property for Equality

8. m/CAD + m/1 = m/DAB 8. Postulate 7 (Protractor) Angle-Addition Assumption

m/ACD + m/2 = m/BCD

9. m/DAB = m/BCD 9. Substitution Principle

10. /DAB > /BCD 10. Definition of Congruent Angles

> > >

> >


9.

1. PQ MN 1. Given

2. /MQP is a right angle; /NQP is a right angle 2. Definition of Perpendicular

3. /MQP > /NQP 3. Theorem 11

4. m/MQP = m/NQP 4. Definition of Congruent Angles

5. RQ bisects /MQP; SQ bisects /NQP 5. Given

6. /MQR > /RQP; /NQS > /SQP 6. Definition of Angle Bisector

7. m/MQR = m/RQP; m/NQS = m/SQP 7. Definition of Congruent Angles

8. m/MQR + m/RQP = m/MQP 8. Postulate 7 (Protractor) Angle-Addition Assumption

m/NQS + m/SQP = m/NQP

9. m/MQR + m/RQP = m/NQS + m/SQP 9. Substitution of Equals

10. m/RQP + m/RQP = m/SQP + m/SQP 10. Substitution of Equals

11. (1 + 1) m/RQP = (1 + 1) m/SQP 11. Distributive Property of Multiplication over Addition

12. 2m/RQP = 2m/SQP 12. Substitution of Equals (1 + 1 = 2)

13. 1/2 • 2m/RQP = 1/2 • 2m/SQP 13. Multiplication Property for Equality

14. 1 • m/RQP = 1 • m/SQP 14. Multiplicative Inverse Property

15. m/RQP = m/SQP 15. Identity Property for Multiplication

16. m/SQP = m/RQP 16. Symmetric Property for Equality

17. m/MQP + m/PQS = m/NQP + m/PQR 17. Addition Property for Equations

18. m/MQP + m/PQS = m/MQS 18. Postulate 7 (Protractor) Angle-Addition Assumption

m/NQP + m/PQR = m/NQR

19. m/MQS = m/NQR 19. Substitution of Equals

20. /MQS > /NQR 20. Definition of Congruent Angles

10.

1. /EQF is a straight angle 1. Definition of Straight Angle - An angle whose sides are opposite rays...

2. m/EQF = 180 2. Definition of Straight Angle - ...giving a measure of 180O

3. m/EQF = m/3 + m/BQF; 3. Postulate 7 (Protractor) Angle-Addition Assumption

m/BQF = m/BQD + m/1

4. m/EQF = m/3 + m/BQD + m/1 4. Substitution of Equals

5. 180 = m/3 + m/BQD + m/1 5. Substitution of Equals

6. AB CD at point Q 6. Given

7. /BQD is a right angle 7. Definition of Perpendicular

8. m/BQD = 90 8. Definition of Right Angle

9. 180 = m/3 + 90 + m/1 9. Substitution of Equals

10. 180 + –90 = m/3 + 90 + m/1 + –90 10. Addition Property for Equals

11. 90 = m/3 + 90 + m/1 + –90 11. Substitution of Equals

12. 90 = m/3 + m/1 + 90 + –90 12. Commutative Property of Addition

13. 90 = m/3 + m/1 + 0 13. Additive Inverse Property

14. 90 = m/3 + m/1 14. Identity Property for Addition

15. /1 and /3 are complementary 15. Definition of Complementary Angles

>

>



p. 264 – Lesson 4 — Theorem 12: If you have straight angles, then those straight angles are congruent.

1. a) Theorem 12 – If you have straight angles, then those straight angles are congruent.

b)

c) Given: /AOB and /CPD are straight angles d) Prove: /AOB > /CPD

e)

1. /AOB and /CPD are straight angles 1. Given

2. m/AOB = 180; m/CPD = 180 2. Definition of Straight Angle

3. m/AOB = m/CPD 3. Substitution of Equals

4. /AOB > /AOB 4. Definition of Congruent Angles

2. m/AOB = m/AOF + m/FOBm/AOB = m/AOD + m/DOBm/AOB = m/AOC + m/COBm/AOB = m/AOE + m/EOB

m/COD = m/COA + m/AODm/COD = m/COF + m/FODm/COD = m/COE + m/EODm/COD = m/COB + m/BOD

m/EOF = m/EOC + m/COFm/EOF = m/EOA + m/AOFm/EOF = m/EOB + m/BOFm/EOF = m/EOD + m/DOF

3.

1. /MON is a straight angle; /RTS is a straight angle 1. Given

2. /MON > /RTS 2. Theorem 12

3. m/MON = m/RTS 3. Definition of Congruent Angles

4. m/MON = m/1 + m/2; m/RTS = m/3 + m/4 4. Postulate 7 (Protractor) - Angle-Addition Assumption


6. m/1 > m/3 6. Given


8. m/1 + m/2 –m/1 = m/3 + m/4 – m/3 8. Subtraction Property for Equations

9. m/1 + m/2 + –m/1 = m/3 + m/4 + –m/3 9. Definition of Subtraction

10. m/1 + –m/1 + m/2 = m/3 + –m/3 + m/4 10. Commutative Property of Addition

11. 0 + m/2 = 0 + m/4 11. Additive Inverse Property

12. m/2 = m/4 12. Identity Property of Addition

13. /2 > /4 13. Definition of Congruent Angles

BA O CO

D

B

P CA

B

O

D

m

C

A

B Z

X

Y

B

A

O

DC

P -1 0 1 2

123


4.

1. /GHI is a straight angle; /XYZ is a straight angle 1. Given

2. /GHI > /XYZ 2. Theorem 12

3. m/GHI = m/XYZ 3. Definition of Congruent Angles

4. m/GHI = m/1 + m/2; m/XYZ = m/3 + m/4 4. Postulate 7 (Protractor) - Angle-Addition Assumption


5. Not necessarily; Not necessarily; we need to know that one pair, either /1 and /3 or /2 and /4, is congruent; 90 degrees.

6.

1. /AQB is a straight angle; /DQC is a straight angle 1. Given

2. /AQB > /DQC 2. Theorem 12

3. m/AQB = m/DQC 3. Definition of Congruent Angles

4. m/DQC = m/3 + m/2; m/AQB = m/2 + m/3 4. Postulate 7 (Protractor) - Angle-Addition Assumption


6. m/1 + m/2 – m/2 = m/2 + m/3 – m/2 6. Subtraction Property for Equations


8. m/1 + m/2 + –m/2 = m/3 + m/2 + –m/2 8. Commutative Property of Addition

9. m/1 + 0 = m/3 + 0 9. Additive Inverse Property

10. m/1 = m/3 10. Identity Property for Addition


7.

1. /MRV is a straight angle; /MNB is a straight angle 1. Given

2. /MRV > /MNB 2. Theorem 12

3. m/MRV = m/MNB 3. Definition of Congruent Angles

4. m/MRV = m/3 + m/TRV; m/MNB = m/2 + m/SNB 4. Postulate 7 (Protractor) - Angle-Addition Assumption

5. m/2 + m/SNB = m/3 + m/TRV 5. Substitution of Equals

6. /2 > /3 6. Given


8. m/2 + m/SNB – m/2 = m/3 + m/TRV – m/3 8. Subtraction Property for Equations

9. m/2 + m/SNB + –m/2 = m/3 + m/TRV + –m/3 9. Definition of Subtraction

10. m/SNB + m/2 + –m/2 = m/TRV + m/3 + –m/3 10. Commutative Property of Addition

11. m/SNB + 0 = m/TRV + 0 11. Additive Inverse Property

12. m/SNB = m/TRV 12. Identity Property for Addition

13. /RNT is a straight angle; /NRS is a straight angle 13. Given

14. /RNT > /NRS 14. Theorem 12

15. m/RNT = m/NRS 15. Definition of Congruent Angles

16. m/RNT = m/4 + m/SNB; m/NRS = m/5 + m/TRV 16. Postulate 7 (Protractor) - Angle-Addition Assumption

17. m/4 + m/SNB = m/5 + m/TRV 17. Substitution of Equals

18. m/4 + m/SNB – m/SNB = m/5 + m/TRV – m/TRV 18. Subtraction Property for Equations

19. m/4 + m/SND + –m/SNB = m/5 + m/TRV + –m/TRV 19. Definition of Subtraction





8.

1. /ASC and /BRC are straight angles 1. Given

2. /ASC > /BRC 2. Theorem 12

3. m/ASC = m/BRC 3. Definition of Congruent Angles

4. m/ASC = m/ASB + m/BSC 4. Postulate 7 (Protractor) - Angle-Addition Assumption

m/BRC = m/BRA + m/ARC

5. m/ASB + m/BSC = m/BRA + m/ARC 5. Substitution of Equals

6. /ASB > /BRA 6. Given

7. m/ASB = m/BRA 7. Definition of Congruent Angles

8. m/ASB + m/BSC – m/ASB = m/BRA + 8. Subtraction Property for Equations

m/ARC – m/BRA

9. m/ASB + m/BSC + –m/ASB = m/BRA + 9. Definition of Subtraction

m/ARC + –m/BRA

10. m/ASB + –m/ASB + m/BSC = m/BRA + 10. Commutative Property of Addition

–m/BRA + m/ARC

11. 0 + m/BSC = 0 + m/ARC 11. Additive Inverse Property

12. m/BSC = m/ARC 12. Identity Property for Addition

13. /BSC > /ARC 13. Definition of Congruent Angles

9.

1. AB XY 1. Given

2. /AXY is a right angle; /BXY is a right angle 2. Definition of Perpendicular

3. /AXY > /BXY 3. Theorem 11

4. m/AXY = m/BXY 4. Definition of Congruent Angles

5. m/AXY = m/1 + m/2; m/BXY = m/3 + m/4 5. Postulate 7 (Protractor) - Angle-Addition Assumption


7. /1 > /2 7. Given


9. XY bisects /WXZ 9. Given

10. /2 > /3 10. Definition of Angles Bisector


12. m/1 = m/3 12. Transitive Property of Equality

13. m/3 + m/3 = m/3 + m/4 13. Substitution (11 and 12 into 6)

14. m/3 = m/3 14. Reflexive Property of Equality

15. m/3 + m/3 + –m/3 = m/3 + m/4 + –m/3 15. Subtraction Property for Equality


17. m/3 + m/3 + –m/3 = m/3 + –m/3 + m/4 17. Commutative Property of Addition

18. m/3 + 0 = 0 + m/4 18. Additive Inverse Property



>


10. a) Yes, the graph has a definite endpoint (starting point) at x = 2/3 and a definite direct (x > 2/3)

b) straight

11. a) 1. /ABC; /ABD2. Point Y3. Point X4. yes5. yes6. yes7. yes, Point B

b) 1. yes2. yes; /ABC, /ACD, or /BCD3. yes4. 180 degrees5. m/ABC = m/ACD = m/BCD6. /ABC > /ACD > /BCD7. no8. cannot judge

cannot judge

c) no; no

Unit III – Fundamental TheoremsPart G — Theorems About Angles – Part 3 (More Than Two Angles)

p. 268 – Lesson 1 — Theorem 13: If two angles are complementary to the same angles or congruent angles, then they are congruent to each other.

1. a) If two angles are complementary to the same angles or congruent angles, then they are congruent to each other.

b)

c) Given: /1 is the complement of /3 d) Prove: /1 > /2/2 is the complement of /3

e)

1. /1 is complementary to /3 1. Given




5. m/1 + m/3 = m/2 + m/3 5. Algebraic Substitution (4 into 2)

6. m/1 + m/3 + –m/3 = m/2 + m/3 + –m/3 6. Addition Property for Equality

7. m/1 + 0 = m/2 + 0 7. Additive Inverse Property (a + –a = 0)

8. m/1 = m/2 8. Identity Property for Addition (a + 0 = a)


3x + 7 9

3x + 7 + -7 9 + -7

3x + 0 2

3x 2

13

3x13

2

1 x2

≥≥≥≥

⋅ ≥ ⋅

⋅ ≥33

x23

≥

yx

BA O CO

C

B

O

D

m

C

A

B Z

X

Y

DC

P -1 0 1 2

123

1

3

2 1

2

4

3

1

1

3E

2

D

BC

A

4 24

2 3

1

3E

D

BC

A

1 2 3 4

C

BA

O

m

,

32

1

O B

C

A O B

C

A

x y = x + 20

x

C A B

4747Part G — Theorems About Angles – Part 3 (More Than Two Angles)

2. a) Theorem 13 - Part 2 - If two angles are complementary to congruent angles, then they are congruent to each other.

b)

c) Given: /1 is complementary to /2 d) Prove: /1 > /4/4 is complementary to /3/2 > /3

e)


/4 is complementary to /3


m/4 + m/3 = 90

3. m/1 + m/2 = m/4 + m/3 3. Substitution of Equals (2 into 2)

4. /2 > /3 4. GIven


6. m/1 + m/2 – m/2 = m/4 + m/3 – m/3 6. Subtraction Property for Equality





3. a) 90 – 38 b) 90 – x c) 90 – 2y52

4. The angles are x and 90 – x.

5. a) b)

x = 90 - x

x + x = 90 - x + x

2x = 90 - 0

2x = 90

12

2x =12

90

1 x =

⋅ ⋅

⋅ 445

x = 45

3y + 5 + 2y = 90

5y + 5 = 90

5y + 5 - 5 = 90 - 5

5y + 0 = 85

5y = 85

1

( )

555y =

15

85

1 y = 17

y = 17

⋅ ⋅

⋅

m ACB = 3y + 5

= 3(17) + 5

= 51+ 5

m ACB = 56

m EDF = 2y

= 2(1

∠

∠

∠77)

m EDF = 34∠

(y - 8) + (3y + 2) = 90

4y - 6 = 90

4y - 6 + 6 = 90 + 6

4y - 0 = 96

4y = 996

14

4y =14

96

1 y = 24

y = 24

⋅ ( )

⋅

⋅

m ACB = y - 8

= 24 - 8

m ACB = 16

m DEF = 3y + 2

= 3(24) + 2

= 72

∠

∠

∠

++ 2

m EDF = 74∠

3

2 1

2

4

3

24

2 3

1

3E

D

BC

A

1 2 3 4

C

BA

O

m

,

32

1

B

C

O B

C

A

x y = x + 20

x 3xO B

C

A

C

O C

A

P D

B

m

BC

A

Y Z

X


c)

6. The angles are x and 90 – x

7. Theorem 13 - If two angles are complementary to congruent angles, then the two angles are congruent to each other.

8.

1. BD bisects /ABC 1. Given

2. /1 > /2 2. Definition of a Bisector of an Angle



4. /4 > /3 4. Theorem 13 - If two angles are complementary to congruent

angles, then the angles are congruent to each other.

9.

1. HP PQ; JQ PQ 1. Given

2. /HPQ is a right angle 2. Definition of Perpendicular

/JQP is a right angle

3. m/HPQ = 90; m/JQP = 90 3. Definition of Right Angle

4. m/HPQ = m/1 + m/2 4. Postulate 7 (Protractor) - Angle-Addition Assumption

m/JQP = m/4 + m/3

5. 90 = m/1 + m/2 5. Substitution of Equals

90 = m/4 + m/3

6. /1 is complementary to /2 6. Definition of Complementary Angles


7. /2 > /3 7. Given



y + (6y -1) = 90

y + 6y -1= 90

y + 6y -1- 90 = 90 - 90

y + 6y

2

2

2

2 -- 91= 0

(y - 7)(y +13) = 0

m ACB = y

= 7

m ACB = 49

m EDF = 6y -1

= 6(7) -1

= 42 -1

m

2

2

∠

∠

∠

∠EEDF = 41

y - 7 = 0Ä or Ä y +13 = 0

y = 7 or y = -13

x =12

90 - x

2 x = 212

90 - x

2x = 1 90 - x

2x = 90 - x

( )

⋅ ⋅ ⋅ ( )⋅ ( )

22x + x = 90 - x + x

3x = 90 - 0

13

3x =13

90

1 x = 30

x = 30

⋅ ⋅

⋅

> >


10.

1. /JML is a right angle 1. Given

2. m/JML = 90 2. Definition of Right Angles

3. m/JML = m/3 + m/4 3. Postulate 7 (Protractor) - Angle-Addition Assumption






11.

1. /ABD is complementary to /BAD 1. Given

/CBE is complementary to /BAD

2. /ABD > /CBE 2. Theorem 13 - If two angles are complementary to the same angle,

then the angles are congruent to each other.

3. m/ABD = m/CBE 3. Definition of Congruent Angles

4. m/DBE = m/EBD 4. Reflexive Property of Equality

5. m/ABD + m/DBE = m/CBE + m/EBD 5. Addition Property for Equality

6. m/ABD + m/DBE = m/ABE 6. Postulate 7 (Protractor) - Angle-Addition Assumption

m/CBE + m/EBD = m/CBD

7. m/ABE = m/CBD 7. Substitution of Equals

8. /ABE > /CBD 8. Definition of Congruent Angles

12.

1. BA CA 1. Given

2. /BAC is a right angle 2. Definition of Perpendicular

3. m/BAC = 90 3. Definition of Right Angle

4. m/1 + m/2 = m/BAC 4. Postulate 7 (Protractor) - Angle-Addition Assumption




8. /2 > /3 8. Theorem 13 - If two angles are supplementary to the same angle,

then the angles are congruent to each other.

>



p. 271 – Lesson 2 — Theorem 14: If two angles are supplementary to the same angle, or congruent angles then they are congruent toeach other.

1. a) Theorem 14 – Part 2 - If two angles are supplementary to the same angle, then they are congruent to each other.

b)

c) Given: /1 is supplementary to /2 d) Prove: /1 > /3/3 is supplementary to /2

e)

1. /1 is supplementary to /2 1. Given

2. m/1 + m/2 = 180 2. Definition of Supplementary Angles



5. m/1 + m/2 = m/3 +m/2 5. Algebraic Substitution

6. m/2 = m/2 6. Reflexive Property of Equality

7. m/1 + m/2 – m/2 = m/3 + m/2 – m/2 7. Subtraction Property of Equality





2. a) Theorem 14 – Part 1 - If two angles are supplementary to congruent angles, then they are congruent to each other.

b)

c) Given: /1 is supplementary to /2 d) Prove: /1 > /4/4 is supplementary to /3/2 > /3

e)


/4 is supplementary to /3


m/4 + m/3 = 180


4. /2 > /3 4. Given


6. m/1 + m/2 – m/2 = m/4 + m/3 – m/3 6. Subtraction Property of Equality





1

3

2 1

2

4

3

1

1

3E

2

D

BC

A

4 24

2 3

1

3E

D

BC

A

1 2 3 4

C

BA

O

m

,

32

1

O B

C

A O B

C

A

x y = x + 20

x y = 3xO B

C

A

O C

B

D

A

O C

A

P D

B

m

,

BC

A

Y Z

X

1

2

4

3

2

3

1

3E

D

B

1 2 3 4

C

BA

O

m

,

32

1

O B

C

A

x y = x + 20

xO B

C

A O C

A

P D

B

A X


3. a) 180 – 42 b) 180 – (x – 3) c) 180 – x2

138 180 – x + 3183 – x

4. The angles are x and 180 – x.

5. a) b)

c)

6. The angles x and 180 – x.

7. Theorem 14 - If two angles are supplements of congruent angles, then they are congruent to each other.

x = 180 - x

x + x = 180 - x + x

2x = 180 - 0

2x = 180

12

2x =12

18⋅ ⋅ 00

1 x = 90

x = 90

⋅

2x + x -15 = 180

2x + x -15 = 180

3x -15 = 180

3x -15 +15 = 1

( )

880 +15

3x - 0 = 195

3x = 195

13

3x =13

195

1 x = 65

x = 65

⋅ ⋅

⋅

m ACB = 2x

= 2 65

m ACB = 130

m EDF = x -15

= 65 -15

m ED

∠( )

∠

∠

∠ FF = 50

x + 12x - 9 = 180

x +12x - 9 = 180

x +12x - 9 -180 = 180 -

2

2

2

( )

1180

x +12x -189 = 0

x - 9 x + 21 = 0

2

( )( )

m ACB = x

= 9

m ACB = 81

m DEF = 12x - 9

= 12 9 - 9

= 108 -

2

2

∠

∠

∠( )

99

m DEF = 99∠

x +16 + 2x -16 = 180

x +16 + 2x -16 = 180

3x = 180

13

3x

( ) ( )

⋅ ==13

180

1 x = 60

x = 60

⋅

⋅

m ACB = x +16

= 60 +16

m ACB = 76

m EDF = 2x -16

= 2 60 -1

∠

∠

∠( ) 66

= 120 -16

m EDF = 104∠

x = 2 180 - x

x = 360 - 2x

x + 2x = 360 - 2x + 2x

3x = 360 + 0

3x =

( )

3360

13

3x =13

360

1 x = 120

x = 120

⋅ ⋅

⋅


8.

1. /ABC and /DCB are straight angles 1. Definition of Straight Angle

2. m/ABC = 180; m/DCB = 180 2. Definition of Straight Angle

3. m/ABC = m/1 + m/2 3. Postulate 7 (Protractor) - Angle-Addition Assumption

m/DCB = m/3 + m/4

4. 180 = m/1 + m/2; 180 = m/3 + m/4 4. Substitution of Equals

5. /1 is the supplement of /2 5. Definition of Supplementary Angles

/3 is the supplement of /4

6. /2 > /3 6. Given

7. /1 > /4 7. Theorem 14 - If two angles are supplements of congruent angles,

then they are congruent to each other.

9.

1. /BFE > /ECA; /CFD > /ECA 1. Given

2. m/BFE = m/ECA; m/CFD = m/ECA 2. Definition of Congruent Angles

3. m/ECA = m/CFD 3. Symmetric Property of Equality

4. m/BFE = m/CFD 4. Transitive Property of Equality

5. /BFE > /CFD 5. Definition of Congruent Angles

6. m/BFE + m/EFC = m/BFC 6. Postulate 7 (Protractor) - Angle-Addition Assumption

m/CFD + m/DFB = m/CFB

7. /BFC is a straight angle 7. Definition of Straight Angle

8. m/BFC = 180; m/BFC = 180 8. Definition of Straight Angle

9. m/BFE + m/EFC = 180 9. Substitution of Equals

m/CED + m/DFB = 180

10. /BFE and /EFC are supplementary angles 10. Definition of Supplementary angles - sum is 180

/CFD and /DFB are supplementary angles

11. /DFB > /EFC 11. Theorem 14 - If two angles are supplements of congruent angles,


10.

1. /ACD is a straight angle 1. Definition of Straight Angle

2. m/ACD = 180 2. Definition of Straight Angle

3. m/ACD = m/2 + m/3 3. Postulate 7 (Protractor) - Angle-Addition Assumption


5. /2 and /3 are supplementary angles 5. Definition of Supplementary Angles

6. /1 is the supplement of /2 6. Given




11.

1. /DBE is a straight angle 1. Definition of Straight Angle

/ECD is a straight angle

2. m/DBE = 180; m/ECD = 180 2. Definition of Straight Angle

3. m/DBE = m/DBA + m/ABE 3. Postulate 7 (Protractor) - Angle-Addition Assumption

m/ECD = m/ECA + m/ACD

4. 180 = m/DBA + m/ABE; 180 = m/ECA + m/ACD 4. Substitution of Equals

5. /DBA and /ABE are supplementary angles 5. Definition of Supplementary Angles

/ECA and /ACD are supplementary angles

6. /ABE > /ACD 6. Given

7. /DBA > /ECA 7. Theorem 14 - If two angles are supplements of congruent angles,


8. m/DBA = m/ECA 8. Definition of Congruent Angles

9. /FCA is a straight angle 9. Definition of Straight Angle

10. m/FCA = 180 10. Definition of Straight Angle

11. m/FCA = m/FCE + m/ECA 11. Postulate 7 (Protractor) - Angle-Addition Assumption

12. 180 = m/FCE + m/ECA 12. Substitution of Equals

13. 180 = m/FCE + m/DBA 13. Substitution of Equals

14. /DBA is a supplement of /FCE 14. Definition of Supplementary Angles

12.

1. m/EGB + m/BGF = 180 1. Given

m/EHD + m/BGF = 180

2. /EGB and /BGF are supplementary angles 2. Definition of Supplementary Angles

/EHD and /BGF are supplementary angles

3. /EGB > /EHD 3. Theorem 14 - If two angles are supplements of congruent angles,


4. m/EGB = m/EHD 4. Definition of Congruent Angles


p. 274 – Lesson 3 — Theorem 15: If two lines intersect, then the vertical angles formed are congruent.

1. a) Theorem 15 – Part 1 - If two lines intersect, then the vertical angles formed are congruent.

b)

c) Given: AB and CD intersect at point E d) Prove: /1 > /3

1

3

2 1

2

4

3

1

1

3E

2

D

BC

A

4 24

2 3

1

3E

D

BC

A

1 2 3 4

C

BA

O

m

,

32

1

O B

C

A O B

C

A

x y = x + 20

x y = 3xO B

C

A O C

A

P D

B


e)

1. AB and CD intersect at point E 1. Given

2. ED and EC are opposite rays 2. Definition of Opposite Rays

EB and EA are opposite rays

3. /1 is adjacent to /2 3. Definition of Adjacent Angles

/3 is adjacent to /2

4. /1 is supplementary to /2 4. Theorem 10 - If the exterior sides of two adjacent angles are

/3 is supplementary to /2 opposite rays, then the angles are supplementary



2. a) Theorem 15 – Part 2 - If two lines intersect, then the vertical angles formed are congruent.

b)

c) Given: AB and CD intersect at point E d) Prove: /2 > /4

e)

1. AB and CD intersect at point E 1. Given

2. ED and EC are opposite rays 2. Definition of Opposite Rays

EB and EA are opposite rays

3. /4 is adjacent to /3 3. Definition of Adjacent Angles

/2 is adjacent to /3

4. /4 is supplementary to /3 4. Theorem 10 - If the exterior sides of two adjacent angles are

/2 is supplementary to /3 opposite rays, then the angles are supplementary



3. a) /3 and /4 are vertical angles; m/3 = 110. Therefore, m/4 = 110

b) /1 and /3 are supplementary

m/1 + m/3 = 180

m/1 + 110 = 180

m/1 = 70

c) /2 and /1 are vertical angles; m/1 = 70. Therefore, m/2 = 70

1

3

2 1

2

4

3

1

1

3E

2

D

BC

A

4 24

2 3

1

3E

D

BC

A

1 2 3 4

C

BA

O

m

,

32

1

O B

C

A O B

C

A

x y = x + 20

x y = 3xO B

C

A

O C

B

D

A

O C

A

P D

B

m

,

BC

A

Y Z

X


4. a) b) /1 and /2 are vertical angles

m/1 = 60

therefore, m/2 = 60

c) m/3 = 120 (part a)

d) /3 and /4 are vertical angles

m/3 = 120

therefore, m/4 = 120

5. Two angles are congruent and supplementary. Each angle measures 90. Therefore, line m line n.

6. Two angles are congruent and complementary. Each angle measures 45. Therefore, m/4 = 45

7. a) increase by 10 b) decreased by 10 c) decrease by 10

8.

9.

10.

11.

m m

m m

m m

∠ + ∠ =

⋅ ∠ + ∠ =

⋅ ∠ + ∠ =

1 3 180

1

23 3 180

21

23 3 180

∠ + ∠ =⋅ ∠ =

⋅ ∠ = ⋅

⋅ ∠

m m

m

m

m

3 2 3 360

3 3 360

1

33 3

1

3360

1 33 120

3 120

11

23

11

2120

1 60

=∠ =

∠ = ⋅ ∠

∠ = ⋅

∠ =

m

m m

m

m

5 20 4 15

5 4 20 20 4 4 15 20

0 0 35

x x

x x x x

x

x

− = +− − + = − + +

+ = +== 35

5 20 155

4 15 155

25

x m APX

x m BPY

m APB

m XPY

− = = ∠+ = = ∠

∠ =∠ == 25

7 35 3 85

7 3 35 35 3 3 85 35

4 50

1

44

x x

x x x x

x

+ = +− + − = − + −

=

⋅ xx

x

x or

= ⋅

⋅ =

=

1

450

150

425

212

1

2

7 35 122 5

3 85 122 5

57

x m NQT

x m RQM

m NQP

+ = = ∠+ = = ∠

∠ =

.

.

.55

57 5m TQM∠ = .

4 36 64

4 100

1

44

1

4100

1100

425

x

x

x

x

x

= +=

⋅ = ⋅

⋅ =

=

4 180

80

80

64

3

x m QPR

m QPR

m MPU

m UPT

m TPR

+ ∠ =∠ =∠ =∠ =∠ = 66

x x

x x

x x

2

2

6 9

6 9 0

3 3 0

= −

− + =−( ) −( ) =

x or x

x x

− = − == =

3 0 3 0

3 3

x m XPW

x m YPZ

m WPZ

m XPY

2 9

6 9 9

171

171

= = ∠− = = ∠

∠ =∠ =

>


12.

1. /1 > /2 1. Given

2. /2 > /4 2. Theorem 15 - If two lines intersect, then the vertical angles

formed are congruent

3. /3 > /4 3. Given

4. m/1 = m/2; m/2 = m/4; m/3 = m/4 4. Definition of Congruent Angles

5. m/4 = m/3 5. Symmetric Property for Equality

6. m/1 = m/3 6. Transitive Property for Equality


13.

1. /APR > /NPB 1. Given

2. /NPB > /MPA 2. Theorem 15 - If two lines intersect, then the vertical angles


3. m/APR = m/NPB 3. Definition of Congruent Angles

m/NPB = m/MPA

4. m/APR = m/MPA 4. Transitive Property of Equality

5. m/MPA = m/APR 5. Symmetric Property of Equality

6. /MPA > /APR 6. Definition of Congruent Angles

14.

1. /MPR > /NPQ 1. Theorem 15 - If two lines intersect, then the vertical angles


2. m/MPR = m/NPQ 2. Definition of Congruent Angles

3. m/MPR = m/MPA + m/APR 3. Postulate 7 (Protractor) - Angle-Addition Assumption

m/NPQ = m/NPB + m/BPQ

4. m/MPA + m/APR = m/NPB + m/BPQ 4. Substitution of Equals (2 into 2)

5. /APR > /NPB 5. Given

6. m/APR = m/NPB 6. Definition of Congruent Angles

7. m/MPA + m/APR – m/APR = m/NPB + 7. Subtraction Property for Equality

m/BPQ – m/NPB

8. m/MPA + m/APR + –m/APR = m/NPB + 8. Definition of Subtraction

m/BPQ + –m/NPB

9. m/MPA + m/APR + –m/APR =m/BPQ + 9. Commutative Property of Addition

m/NPB + –m/NPB

10. m/MPA + 0 = m/BPQ + 0 10. Additive Inverse Property

11. m/MPA = m/BPQ 11. Identity Property for Addition

12. /MPA > /QPB 12. Definition of Congruent Angles


15.

1. PA bisects /MPR 1. Given

2. /APR > /APM 2. Definition of Angle Bisector

3. /APR > /BPQ; /APM > /BPN 3. Theorem 15 - If two lines intersect, then the vertical angles


4. m/APR = m/APM; m/APM = m/BPN 4. Definition of Congruent Angles

5. m/APR = m/BPN 5. Transitive Property of Equality

6. m/APR = m/BPQ 6. Definition of Congruent Angles

7. m/BPN = m/APR 7. Symmetric Property of Equality

8. m/BPN = m/BPQ 8. Transitive Property of Equality

9. /BPN > /BPQ 9. Definition of Congruent Angles

10. PB is between PQ and PN so that 10. Definition of Betweeness for Rays

m/NPB + m/BPQ = m/NPQ

11. PB bisects /NPQ 11. Definition of Angle Bisector

Unit III — Fundamental TheoremsPart H — Theorems about Parallel Lines

p. 277 – Lesson 1 — Postulate 11– Corresponding Angles of Parallel Lines: If two parallel lines are cut by a transversal, then corresponding angles are congruent.

1. Postulate 11 - If two parallel lines are cut by a transversal, then corresponding angles are congruent.

2. a) Yes b) No c) No d) Yes

3. a) ,1 b) ,1 c) ,1 d) ,2,2 ,2 ,3 ,3,3 ,3 ,4 ,4,4 ,5

4. a) 1. /2, /3, /6, /7 b) 1. /3, /4, /5, /6

2. /1, /4, /5, /8 2. /1, /2, /7, /8

3. /1, /2, /3, /4 or /5, /6, /7, /8 3. /2, /4, /6, /8 or /1, /3, /5, /7

5. a) /2 and /3 b) /1 and /5 c) /3 and /6 d) /2 and /7

/1 and /4 /3 and /7 /4 and /5 /1 and /8

/8 and /5 /2 and /6

/6 and /7 /4 and /8

6. a) Alternate Interior f) Alternate Exterior

b) Corresponding g) Interior Angles on the

c) Alternate Exterior same side of the transversal

d) Interior Angles on the h) Corresponding

same side of the transversal i) Vertical

e) Alternate Interior

7. a) No Yes No

b) No No No

c) No Yes Yes

d) No No Yes

e) No No Yes

f) No Yes Yes

8. Statement Reason

1. BC EF 1. Given

BA ED

2. /B > /DPC; /DPC > /E 2. Postulate 11 - If two parallel lines are cut by a transversal,then corresponding angles are congruent.

3. m/B 5 m/DPC 3. Definition of congruent angles.m/DPC 5 m/E

4. m/B 5 m/E 4. Transitive Property of Equality

5. /B > /E 5. Definition of Congruent Angles

1 2

3 4

5 6

7 8

<1

<2

t

/1 > /5

/3 > /7

/2 > /6

/4 > /8


Unit III — Fundamental TheoremsPart H — Theorems About Parallel Lines

p. 282 – Lesson 2 — Theorem 16: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

1. a) Theorem 16 - If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

b)

c) Given: Line , is parallel to line m cut by transversal t.d) Prove: Alternate Interior angles are congruent.

e) Statement Reason

1. , || m, cut by transversal t 1. Given

2. /3 > /7 2. Postulate 11 - If two parallel lines are cut by a transversal,

then corresponding angles are congruent.


4. /6 > /7 4. Theorem 15 - If two lines intersect, then the vertical angles

formed are congruent.


6. m/6 = m/3 6. Substitution (3 into 5)


2.

Statement Reason

1. , || m ; , and m are cut by transversal t 1. Given

2. /3 and /4 form a Linear Pair 2. Definition of Linear Pair - Two angles which have a common

/5 and /6 form a Linear Pair side (they are adjacent), and whose exterior sides are

opposite rays.

3. /3 and /4 form a Straight Angle 3. Definition of Straight Angle - Sides are opposite rays, giving

/5 and /6 form a Straight Angle a measure of 180 degrees.

4. m/3 + m/4 = 180 4. Definition of Straight Angle.

m/5 + m/6 = 180


/5 and /6 are supplementary angles

6. /6 > /3 6. Part I of Theorem 16 - If two parallel lines are cut by a transversal,

then alternate interior angles are congruent.

7. /5 > /4 7. Theorem 14 - If two angles are supplementary to congruent

angles, then they are congruent to each other.

1 2

3 4

5 6

7 8

<

m

t

1 2

3 4

5 6

7 8

<

m

t


61Part H – Theorems About Parallel Lines

3. Corollary 16a - If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

c) Given: Line , || m; , and m are cut by a transversal t.d) Prove: Alternate Exterior Angles are Congruent.

e) Statement Reason

1. , || m; , and m are cut by a transversal t. 1. Given

2. /3 > /6 2. Theorem 16 - If two parallel lines are cut by a transversal,


3. /2 > /3 3. Theorem 15 - If two lines intersect, then the vertical

/6 > /7 angles formed are congruent.

4. m/2 = m/3 4. Definition of Congruent Angles.

m/3 = m/6

m/6 = m/7

5. m/1 = m/2 5. Transitive Property of Equality.


4.

Statement Reason

1. , || m ; , and m are cut by transversal t 1. Given

2. /1 and /2 form a Linear Pair 2. Definition of Linear Pair - Two Angles which have a common

/7 and /8 form a Linear Pair side (they are adjacent), and whose exterior sides are

opposite rays.

3. /1 and /2 form a Straight Angle 3. Definition of Straight Angle - Sides are opposite rays, giving

/7 and /8 form a Straight Angle a measure of 180 degrees.

4. m/1 + m/2 = 180 4. Definition of Straight Angle.

m/7 + m/8 = 180


/7 and /8 are supplementary angles

6. /2 > /7 6. Part I of Corollary 16a - If two parallel lines are cut by a transversal,

then alternate exterior angles are congruent.

7. /1 > /8 7. Theorem 14 - If two angles are supplementary to congruent angles,

then they are congruent to each other

1 2

3 4

5 6

7 8

<

m

t

1 2

3 4

5 6

7 8

<

m

t

5. m/3 = 100 - 55 = 45

6. 4x - 40 = x + 20 m/ABD = 4x -40

4x - 40 + 40 = x + 20 + 40 = 4(20 - 40

4x = x + 60 = 80 - 40

4x - x = x + 60 - x m/ABD = 40

3x = 60

1/3 • 3 x = 1/3 • 60x = 20

7. a) 75 d) 105

b) 75 e) 105

c) 75 f) 105

8. a) Corresponding Angles b) Alternate Interior Angles

m/1 = m/6 m/4 = m/6

3x + 7 = 5x - 3 8x + 12 = 2x + 54

3x + 7 - 3x = 5x - 3 - 3x 8x - 2x + 12 = 2x + 54 - 2x

7 = 2x - 3 6x + 12 = 54

7 + 3 = 2x - 3 + 3 6x + 12 - 12 + 54 - 12

10 = 2x 6x = 42

1/2 • 10 = 1/2 • 2x 1/6 • 6x = 1/6 • 42

5 = x x = 7

c) Corresponding Angles

m/4 = m/8

x2 + 5x = 9x + 12

x2 + 5x - 9x - 12 = 9x + 12 - 9x - 12

x2 - 4x - 12 = 0

(x - 6) (x + 2) = 0

x - 6 = 0 x + 2 + 0

x = 6 x = -2 This answer creates an angle which would measure -6 degrees.

At this point in our study, we do not use negatives for angles.

9. Corresponding Angles Congruent

1. /ABE > /ACD /ABE , /EBD, and /DBC form a straight angles.

m/ABE = m/ACD m/ABE + m/EBD + m/DBC = 180

x = 57º x + y + 63 = 180

57 + y + 63 = 180

y + 120 = 180

y + 120 - 120 = 180 -120

y = 60

10. /AED and /ABC are Congruent Corresponding Angles

/AED is a right angle. So, m/AED = 90.

Therefore, m/ABC = 90.

Since m/ABD + m/DBC = m/ABC,

51 + m/DBC = 90

So m/DBC = 90 - 51

= 39 = x



10. Continued

/DBC and /BDE are Congruent Alternate Interior Angles

Thus, m/BDE = x = 39

/BDE, /BDC, and /CDF form a straight angle

m/BDE + m/BDC + m/CDF = 180

39 + m/BDC + 68 = 180

m/BDC + 107 = 180

m/BDC + 107 = 180

m/BDC = 73 = z

/EDC and /DCG are Congruent Alternate Interior Angles.

m/EDC = m/EDB + m/BDC

= x + m/BDC

= 39 + 73

m/EDC = 112º

Therefore, m/DCG = 112 = y

11.

/ACD and m/ABE are Congruent Corresponding Angles.

(5x + y) + (5x - y) = 80

10x = 80

1/10 • 10 x = 1/10 • 80x = 8

/BEC and /ECD are Congruent Alternate Interior Angles.

2x + y = 5x - y m/ACE = 5x + y

2x + y + y = 5x - y + y = 5 (8) + 12

2x + 2y = 5x = 40 + 12

2x + 2y = 5x m/ACE = 52

2y = 3x

2y = 3 • 8 m/BEC = 2x + y

2y = 24 2 (8) + 12

1/2 • 2y = 1/2 • 24 = 16 + 12

y = 12 m/BEC = 28

12.

Statement Reason

1. p || q, s || t 1. Given

2. /GHC > /IBD 2. Theorem 16 - If two parallel lines are cut by a transversal,

/IBD > /AEF then alternate interior angles are congruent.

3. m/GHC = m/IBD 3. Definition of Congruent Angles.

m/IBD = m/AEF

4. m/GHC = m/AEF 4. Transitive Property of Equality

5. /GHC > /AEF 5. Definition of Congruent Angles.



1. , || m, p || q 1. Given



3. /4 > /9 3. Theorem 16 - If two parallel lines are cut by a transversal,



m/4 = m/9

5. m/2 = m/9 5. Transitive Property of Equality.

6. /2 > /9 6. Definition of Congruent Angles.


1. DE || BC 1. Given

2. /2 > /AED 2. Postulate 11 - If two parallel lines are cut by a transversal,


3. /1 > /DEB 3. Theorem 16 - If two parallel lines are cut by a transversal,


4. m/2 = m/AED 4. Definition of Congruent Angles.

m/1 = m/DEB

5. m/1 = m/2 5. Given

6. m/AED = m/DEB 6. Substitution of Equals.

7. /AED > /DEB 7. Definition of Congruent Angles.

8. DE is Between DA and EB 8. Definition of Betweenness (Rays - Page 131).

9. m/AED + m/DEB = m/AEB 9. Postulate 7 - Protractor - Fourth Assumption - Angle

Addition Assumption.

10. ED Bisects /AEB 10. Definition of Angle Bisector.


1. AC Bisects /DAB 1. Given

2. /DAN > /CAB 2. Definition of Angle Bisector.

3. EN || AB 3. Given

4. /CAB > /ENA 4. Theorem 16 - If two parallel lines are cut by a transversal,


5. m/DAN = m/CAB 5. Definition of Congruent Angles.

m/CAB = m/ENA

6. m/DAN = m/ENA 6. Transitive Property of Equality.

7. /DAN > /ENA 7. Definition of Congruent Angles.


p. 285 – Lesson 3 — Theorem 17: If two parallel lines are cut by a transversal, then interior angles on the same side

of the transversal are supplementary

1. a) Theorem 17 - If two parallel lines are cut by a transversal, then interior angles on the same side


b)

c) Given: Line , is parallel to line m cut by transversal t.

d) Prove: Interior Angles on the same side of the transversal are supplementary.

e) Statement Reason

1. , || m ; , and m are cut by transversal t. 1. Given

2. /6 > /2 2. Postulae 11 - If two parallel lines are cut by a transversal,



4. /2 and /4 are a linear pair 4. Definition of Linear Pair - exterior sides of angles

are opposite rays.

5. /2 and /4 are supplementary 5. Definition of Supplementary Angles. (Linear Pair).

6. m/2 + m/4 = 180 6. Definition of Supplementary Angles (Sum is 180).

7. m/6 + m/4 = 180 7. Substitution (3 into 6).

8. /6 and /4 are supplementary 8. Definition of Supplemenary Angles.

2. Statement Reason

1. , || m; , and mare cut by transversal t. 1. Given




4. /3 and /1 are a linear pair 4. Definition of Linear Pair - Exterior sides of angles

are opposite rays.

5. /3 and /1 form a straight angle 5. Definition of straight Angles - an angle whose sides

are opposite rays.

6. m/3 + m/1 = 180 6. Definition of Straight Angle - Giving a measure of 180.

7. m/3 + m/5 = 180 7. Substitution (3 into 7).

8. /3 and /5 are supplementary angles 8. Definition of Supplemenary Angles.

3. Theorem 17 - Corollary - If two parallel lines are cut by a transversal, then exterior angles on the same side


1 2

3 4

5 6

7 8

<

m

t



3. Continued

Given: Line , is parallel to line m cut by transversal t.

Prove: Exterior Angles on the same side of the transversal are supplementary.

Statement Reason

1. , || m ; , and m are cut by transversal t. 1. Given

2. Angles 4 and 6 are Supplementary Angles 2. Theorem 17 - If two parallel lines are cut by a transversal,

then interior angles on the same side of the transversal

are supplementary.

3. m/4 + m/6 = 180 3. Definition of Supplementary Angles.

4. /1 > /4 ; /7 > /6 4. Theorem 15 - If two lines intersect, then the vertical

angles formed are congruent.

5. m/1 = m/4 ; m/7 = m/6 5. Definition of Congruent Angles.

6. m/1 + m/7 = 180 6. Substitution of Equals (5 into 3).

7. /1 and /7 are Supplementary Angles 7. Definition of Supplementary Angles.

4. Statement Reason

1. , || m; , and m are cut by transversal t. 1. Given

2. Angles 3 and 5 are Supplementary Angles 2. Theorem 17 - If two parallel lines are cut by a transversal,


are supplementary.

3. m/3 + m/5 = 180 3. Definition of Supplementary Angles.

4. /2 > /3 ; /8 > /5 4. Theorem 15 - If two lines intersect, then the vertical


5. m/2 = m/3 ; m/8 = m/5 5. Definition of Congruent Angles.

6. m/2 + m/8 = 180 6. Substitution of Equals (5 into 3).

7. /2 and /8 are Supplementary Angles 7. Definition of Supplementary Angles.

5.

/ABC and /DCB are Interior Angles on the same side of Transversal BC of two parallel lines. /ABC is a right angle.

m/ABC + m/DCB = 180

90 + 3x = 180

90 + 3x - 90 = 180 - 90

3x = 90

1/3 • 3 x = 1/3 • 90

x = 30

/BAD and /CDA are Interior Angles on the same side of Transversal AD of two parallel lines

1 2

3 4

5 6

7 8

<

m

t

5. Continued

m/BAD + m/CDA = 180

y - 10 + 100 = 180

y + 90 = 180

y + 90 - 90 = 180 - 90

y = 90

6.

/MNP and /NPQ are Interior Angles on the same side of Transversal NP of two parallel lines.

m/MNP + m/NPQ = 180

m/MNP + 132 = 180

m/MNP + 132 - 132 = 180 - 132

m/MNP = 48

m/1 = 1/2 m/MNP 1/2 • 48 = 24

m/2 = 1/2 m/MNP 1/2 • 48 = 24

/NRP > /4 - Alternate Interior Angles

m/4 = 1/2 • m/NPQ = 1/2 • 132 = 66. Therefore, m/NRP = 66

7. m/5 = 100 Theorem 17 - Interior Angles Supplementary

8. m/5 = 73 Theorem 16 - Alternate Interior Angles Congruent

9. m/5 = 104 Postulate 11 - Corresponding Angles Congruent

10. m/5 = 76 Theorem 15 - Vertical Angles Congruent

11. m/5 = 82 /5 > /4 - Alternate Interior Angles Congruent

/4 is supplementary to /3 - Linear Pair

12. m/5 = 120 Theorem 17 - Interior Angles Supplementary

m/5 + m/2 = 180

2 • m/2 + m/2 = 180

3 • m/2 = 180

m/2 = 60

m/5 = 2 • m/2 = 2 • 60 = 120


1. m || n 1. Given

2. /ABG is Supplementary to /HED 2. Theorem 17 - If two parallel lines are cut by a transversal,


are supplementary.

3. m/ABG + m/HED = 180 3. Definition of Supplementary Angles.

4. /HED > /FEG 4. Theorem 15 - If two lines intersect, then the vertical


5. m/HED = m/FEG 5. Definition of Congruent Angles.

6. m/ABG + m/FEG = 180 6. Substitution of Equals (5 into 3).

7. /ABG is supplementary to /FEG 7. Definition of Supplementary Angles.




1. AB || CD 1. Given

2. /BAC and /DCA are Supplementary Angles 2. Theorem 17 - If two parallel lines are cut by a transversal,


are supplementary.

3. m/BAC + m/DCA = 180 3. Definition of Supplementary Angles.

4. m/BAC = m/BAE + m/1 4. Postulate 7 - Protractor - Fourth Assumption -

m/DCA = m/DCE + m/2 “Angle Addition” Postulate.

5. m/BAE + m/1 + m/DCE + m/2 = 180 5. Substitution (5 into 3).

6. m/1 and m/2 are Complementary Angles 6. Given

7. m/1 + m/2 = 90 7. Definition of Complementary Angles.

8. m/BAE + m/1 + m/DCE + m/2 8. Subtraction Property of Equality ( 5-7).

- ( m/1 + m/2) = 180 - 90

9. m/BAE + m/1 + m/DCE + m/2 + 9. Definition of Subtraction (a - b means a + - b)

- (m/1 + m/2) = 180 - 90

10. m/BAE + m/1 + m/DCE + m/2 + 10. Definition of Opposite (- (a) means - 1 • a)

- 1 3 (m/1 + m/2) = 180 - 90

11. m/BAE + m/1 + m/DCE + m/2 + 11. Distributive Property of Multiplication- 1 3 m/1 - 1x m/2 = 180 - 90 Over Additon.

12. m/BAE + m/1 + m/DCE + m/1 + 12. Property of (- 1) for Multiplication (-1 • a = - a).

- m/1 - m/2 = 180 - 90

13. m/BAE + m/DCE + m/1 + - m/1 + 13. Commutative Property of Addition (twice).

+ m/2 + - m/2) = 180 - 90

14. m/BAE + m/DCE + 0 + 0 = 180 - 90 14. Additive Inverse Property ( a + - a = 0) - (twice).

15. m/BAE + m/DCE = 180 - 90 15. Identity Property for Addition (a + 0 = a) - (twice).

16. m/BAE + m/DCE = 90 16. Substitution ( 180 - 90 = 90 )

17. m/BAE and /DCE are Complementary Angles 17. Definition of Complementary Angles.

18. AE bisects /BAC 18. Given

19. /BAE > /1 19. Definition of Angle Bisector.

20. m/BAE = m/1 20. Definition of Congruent Angles.

21. m/BAE + m/2 = 90 21. Substitution ( 20 into 7).

22. /BAE and /2 are Complementary Angles 22. Definition of Complementary Angles.

23. /DCE > /2 23. Theorem 13 - If two angles are complementary to the

same angle, then they are congruent to each other.

24. CE is Between CA and CD 24. Definition of Betweeness for Rays ( or segments) - CA,

CD, and CE are co-planar, and A < E < D or D < E < A

on the three segments.

25. CE bisects /DCA 25. Definition of Angle Bisector - A ray (segment) which is

between the sides of an angle, and divides the angle

into two congruent angles.


1. EF || CB 1. Given

2. /EFD and /FDC are Supplementary Angles. 2. Theorem 17 - If two parallel lines are cut by a transversal,


are supplementary.

3. m/EFD + m/FDC = 180 3. Definition of Supplementary Angles.

4. FD || AC 4. Given

5. /AEF is congruent to /EFD 5. Theorem 16 - If two parallel lines are cut by a transversal,


6. m/AEF = m/EFD 6. Definiton of Congruent Angles.

7. m/AEF + m/FDC = 180 7. Substitution ( 6 into 3 )

8. /AEF is Supplementary to /FDC 8. Definition of Supplementary Angles.


p. 287 – Lesson 4 — Theorem 18: If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

1. a) Theorem 18 - If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

b)

c) Given: Line , is parallel to line m.

Transversal t is perpendicular to line ,d) Prove: Transversal t is perpendicular to line m.

e) Statement Reason

1. , || m ; t ,1. Given

2. /1 is a right angle. 2. Definition of Perpendicular Lines

3. m/1 = 90 3. Definition of a Right Angle.




6. 90 = m/5 6. Substitution of Equals - ( 3 into 5 )

7. /5 is a right angle 7. Definition of Right Angle.

8. Transversal t is perpendicular to line m 8. Definition of Perpendicular Lines.

2. m/AEK = 87º - contradicts the other information. All angles in this diagram must be right angles.

3. m/DAE = 43 , m/EAB = 47 contradicts AE Bisects /DAB . AE Bisects /DAB means /DAE /EAB.

4. No contradictory information

5. /JHM and /DLM cannot be supplementary since m/ JHM must be 90 and we do not know what m/DLM is. AE || GM,

CJ AF means CJ is also perpendicular to GM.

1 2

3 4

5 6

7 8

<

m

t



6. a) 90 ; Theorem 18 - If a given line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

b) 30 , Theorem 16 - If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

c) 60 ; /DBA , /ABC , and /CBE form a straight angle

90 + 30 + m/CBE = 180

m/CBE = 60

d) 60 ; Postulate 11 ; If two parallel lines are cut by a transversal, then corresponding angles are congruent.

e) 60 ; /CAB and /BAD are complementary.

7. Statement Reason

1. AB || DC ; DA AB 1. Given

2. DA DC 2. Theorem 18 - If a given line is perpendicular to one of

two parallel lines, then it is perpendicular to the other.

3. /DAB is a right angle 3. Definition of Perpendicular.

/ADC is a right angle

4. /DAB > /ADC 4. Theorem 11 - If you have right angles, then those right

angles are congruent.

5. m/DAB = m/ADC 5. Definition of Congruent Angles.

6. /AXZ > /DZX 6. Theorem 16 - If two parallel lines are cut by a transversal,


7. m/AXZ = m/DZX 7. Definition of Congruent Angles.

8. m/DYX = m/DAB + m/AXZ 8. Given

9. m/DYX = m/ADC + m/DZX 9. Substitution ( 5 and 7 into 8 )

8. Statement Reason

1. QR || MN ; PM MN 1. Given

2. PM QR 2. Theorem 18 - If a given line is perpendicular to one of


3. m/PQR is a right angle 3. Definition of Perpendicular.

m/PMN is a right angle

4. m/PQR = 90 4. Definition of a Right Angle.

m/PMN = 90

5. m/PQR + m/PMN = 90 + 90 5. Addition Property for Equality.

6. m/PQR + m/PMN = 180 6. Substitution of Equals ( 90 + 90 = 180 ).

7. /PQR and /PMN are supplementary 7. Definition of Supplementary Angles

9. Statement Reason

1. PC || QD 1. Given

AB QD

2. AB PC 2. Theorem 18 - If a given line is perpendicular to one of


3. /CPB is a right angle 3. Definition of Perpendicular.

4. m/CPB = 90 4. Definition of a Right Angle.

9. continued Statement Reason

5. PR bisects /CPB 5. Given

6. /2 > /1 6. Definition of Angle Bisector


8. m/1 + m/2 = m/CPB 8. Postulate 7 - Protractor - Fourth Assumption -

“Angle Additon” Assumption.

9. m/1 + m/1 = m/CPB 9. Subsitution ( 7 into 8)

10. ( 1 + 1 ) • m/1 = m/CPB 10. Distributive Property of Multiplication over Addition.

11. 2 • m/1 = m/CPB 11. Substitution ( 1 + 1 = 2).

12. 2m/1 = 90 12. Substitution ( 4 into 11 ).

13. 1/2 • 2 m/1 = 1/2 • 90 13. Multiplication Property for Equality.

14. 1 • m/1 = 1/2 • 90 14. Multiplicative Inverse Property.

15. m/1 = 1/2 • 90 15. Identity Property for Multiplication.

16. m/1 = 45 16. Substitution ( 1/2 • 90 = 45 )

10. a) 72 f) 18

b) 90 g) 72

c) 72 h) 108

d) 18 i) 72

e) 90 j) 108


1. RU || SV, RS || UT 1. Given

RS RU

2. RS SW 2. Theorem 18 - If a given line is perpendicular to one of


3. /RSW is a right angle 3. Definition of Perpendicular Lines.

4. m/RSW = 90 4. Definition of Right Angle.

5. m/RSU + m/USW = m/RSW 5. Postulate 7 - Protractor - Fourth Assumption -

“Angle Additon” Assumption.

6. m/USW = 55 6. Given

7. m/RSU + 55 = m/RSW 7. Subsitution ( 6 into 5 ).

8. m/RSU + 55 = 90 8. Subsitution ( 4 into 7 ).

9. m/RSU + 55 + - 55 = 90 + - 55 9. Additon Property for Equality.

10. m/RSU + 0 = 90 + - 55 10. Additive Inverse Property.

11. m/RSU = 90 + - 55 11. Identity Property of Addition

12. m/RSU = 35 12. Substitution ( 90 + - 55 = 35 ))

13. /RSU > /SUT 13. Theorem 16 - If two parallel lines are cut by a transversal,


14. m/RSU = m/SUT 14. Definition of Congruent Angles.

15. 35 = m/SUT 15. Substitution ( 12 into 14 ).



11. continued Statement Reason

16. TU bisects /SUV 16. Given

17. /SUT > /TUV 17. Definition of Angle Bisector.

18. m/SUT = m/TUV 18. Definition of Congruent Angles.

19. 35 = m/TUV 19. Substitution ( 15 into 18 ).

20. m/TUV = 35 20. Symmetric Property of Equality

Statement Reason

12.

1. AC || UW; AD || UX 1. Given

EY AC

2. EY UW 2. Theorem 18 - If a given line is perpendicular to one of


3. /ABE is a right angle 3. Definition of Perpendicular Lines.

/UVY is a right angle

4. /ABE > /UVY 4. Theorem 11 - If you have right angles, then those right

angles are congruent.

5. m/ABE = m/UVY 5. Definition of Right Angle.

6. /YDA > /EXU 6. Theorem 16 - If two parallel lines are cut by a transversal,


7. m/YDA = m/EXU 7. Definition of Congruent Angles.

8. m/YDA + m/ABE = m/EXU + m/UVY 8. Addition Property of Equality ( 7 + 5 ).


Unit III – Fundamental TheoremsPart H — Theorems About Parallel Lines

p. 291 – Lesson 5 — Theorem 19: If two lines are cut by a transversal so that corresponding angles are congruent,

then the two lines are parallel.

1. a) Theorem 19: If two lines are cut by a transversal so that corresponding angles are congruent, then the two lines are parallel.

b)

c) Given: Line < and line m are cut by transversal t

/1 > /5

d) Prove: < || m

e) Statement Reason

1. Line < and line m are cut by transversal t 1. Given

/1 >/5


3. Suppose < ||y m 3. Indirect Proof Assumption

4. Draw auxiliary line <1 through the intersection 4. Postulate 9 – In a plane, through a point

point of line < and transversal t so that <1 || m. not on a given line, there is exactly one line

parallel to the given line.

5. /11 > /5 5. Postulate 11 – If two parallel lines are cut

by a transversal, then corresponding angles

are congruent.


7. m/11 = m/1 7. Substitution (2 into 6)

8. But m/11 ≠ m/1 8. Postulate 7 – Protractor Second Assumption –

To every pair of rays with a common endpoint,

there corresponds exactly one real number from

0 to 180, inclusive, called the unique measure

of the angle formed by the rays.

9. Our assumption is false, and 9. Reductio ad Absurdum

line < is parallel to line m.

2. a and c 6. a and d 10. no 14. yes

3. b and d 7. b and c 11. yes 15. yes

4. a and c 8. a and d 12. yes 16. no

5. b and d 9. a and c 13. yes 17. yes

1 2

3 4

5 6

7 8

<

m

t

<1


1. BC##$ || EF##$ 1. Given

2. /3 > /2 2. Postulate 11 – If two parallel lines are cut


are congruent.

3. /1 > /3 3. Given

4. m /1 = m /3 4. Definition of Congruent Angles

m /3 = m /2

5. m /1 = m /2 5. Transitive Property of Equality


7. BA##$ || ED##$ 7. Theorem 19 – If two lines are cut by a transversal

so that corresponding angles are congruent, then

the two lines are parallel.


1. BC### || AE##$ 1. Given

2. /1 > /3 2. Theorem 16 – If two parallel lines are cut by

a transversal, then alternate interior angles

are congruent.


4. /3 > /2 4. Given


6. m/1 = m/2 6. Transitive Property of Equality


8. DC### || AB### 8. Theorem 19 – If two lines are cut by a transversal




1. AB || CD 1. Given

2. /BEG > /DFG 2. Postulate 11 – If two parallel lines are cut


are congruent.

3. EH##$ bisects /BEG 3. Given

FJ##$ bisects /DFG

4. /BEH > /HEG 4. Definition of Angle Bisector

/DFJ > /JFG

5. m/BEG = m/DFG 5. Definition of Congruent Angles

m/BEH = m/HEG

m/DFJ = m/JFG

6. m/BEG = m/BEH + m/HEG 6. Postulate 7 – Protractor Fourth Assumption –

m/DFG = m/DFJ + m/JFG

7. m/BEH + m/HEG = m/DFJ + m/JFG 7. Substitution of Equals (6 into 5)



p. 294 – Lesson 6 — Theorem 20: If two lines are cut by a transversal so that alternate interior angles are congruent,


1. a) Theorem 20: If two lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel.

b)


/3 > /6

d) Prove: < || m

e) Statement Reason


2. /3 > /6 2. Given


4. /2 and /3 are vertical angles 4. Definition of Vertical Angles – Two angles

with a common vertex whose sides are

opposite rays.

5. /2 > /3 5. Theorem 15 – If two lines intersect, then

the vertical angles formed are congruent.


7. m/2 = m/6 7. Substitution of Equals (6 into 3)


9. < || m 9. Theorem 19 – If two lines are cut by a transversal



2. a) Corollary 20a: If two lines are cut by a transversal so that alternate external angles are congruent, then the two lines are parallel.

b)

1 2

3 4

5 6

7 8

<

m

t

1 2

3 4

5 6

7 8

<

m

t



2. (continued)


/3 > /6

d) Prove: < || m

e) Statement Reason


2. /1 > /8 2. Given


4. /1 and /4 are vertical angles 4. Definition of Vertical Angles – Two angles

/5 and /8 are vertical angles with a common vertex whose sides are

opposite rays.


/5 > /8 the vertical angles formed are congruent.


m/5 = m/8

7. m/4 = m/5 7. Substitution (6 into 3; 3 into 6)


9. < ||m 9. Theorem 20 – If two lines are cut by a transversal

so that alternate interior angles are congruent,


3. b and c 7. c and d 11. yes 15. yes

4. b and d 8. a and c 12. no 16. yes

5. a and d 9. b and d 13. no 17. yes

6. a and b 10. d and a 14. yes 18. yes


1. /1 > /2 1. Given

2. BC### || FE### 2. Theorem 20 – If two lines (BC### and FE### ) are cut by

a transversal (AD### ) so that alternate interior angles

are congruent, then the two lines are parallel.


1. AF##$ > AD### ; CD##$ > AD### 1. Given

2. /DAF is a right angle 2. Definition of Perpendicular Lines

/ADC is a right angle

3. /DAF > /ADC 3. Theorem 11 – If you have right angles,

then those right angles are congruent

4. AF##$ || CD##$ 4. Theorem 20 – If two lines (AF##$ and DC##$ ) are cut by

a transversal (AD### ) so that alternate interior angles

are congruent, then the two lines are parallel.



p. 297 – Lesson 7 — Theorem 21: If two lines are cut by a transversal so that interior angles on the same side of the transversal are supplementary, then the two lines are parallel.

1. a) Theorem 21: If two lines are cut by a transversal so that interior angles on the same side of the transversal

are supplementary, then the two lines are parallel.

b)



d) Prove: < || m

e) Statement Reason



3. /1 and /3 form a linear pair 3. Definition of Linear Pair – Two angles which have

a common side (they are adjacent), and whose

exterior sides are opposite rays.

4. /1 is supplementary to /3 4. Theorem 10 – If the exterior sides of two adjacent

angles are opposite rays, then the angles are

supplementary.

5. /1 > /5 5. Theorem 14 – If two angles are supplementary

to the same angle, then they are congruent

to each other.




2. a) Corollary 21a: If two lines are cut by a transversal so that exterior angles on the same side of the transversal

are supplementary, then the two lines are parallel.

b)

1 2

3 4

5 6

7 8

<

m

t

1 2

3 4

5 6

7 8

<

m

t

2. (continued)



d) Prove: < || m

e) Statement Reason





/8 > /5 the vertical angles formed are congruent.


m/8 = m/5

6. m/3 + m/5 = 180 6. Substitution (5 into 3)

7. /3 is supplementary to /5 7. Definition of Supplementary Angles


so that interior angles on the same side of the

transversal are supplementary, then the two lines

are parallel.

3. a and c 5. b and d 7. c and d 9. b and c

4. b and d 6. c and d 8. a and d 10. a and d

11. 2x + 4x = 180

6x = 1801/6 • 6x = 1/6 • 180

x = 30

12. 2x + y + 120 = 180

+2x – y + 140 = 180

4x +0 •y + 260 = 360

4x + 260 – 260 = 360 – 260

4x = 1001/4 • 4x = 1/4 • 100

x = 25

2(25) + y + 120 = 180

y + 170 = 180

y + 170 – 170 = 180 – 170

y = 10

13. If AB### is to remain parallel to DC### , then

m/2 must decrease by 30 degrees.

14. m /1 +m /2 = 180

x2 + 8x + 4x + 20 = 180

x2 + 12x + 20 = 180

x2 + 12x + 20 – 180 = 180 – 180

x2 + 12x – 160 = 0

(x + 20) (x – 8) = 0

x + 20 = 0 or x – 8 = 0

x = –20 or x = 8

If x = 8, then m /2 = 4 (8) + 20 = 52, and

m /1 = (8)2 + 8 (8) = 64 + 64 = 128. Since

128 + 52 =180, x = 8 is an acceptable answer.

15. yes

16. no

17. yes

18. yes




1. /R and /S are supplementary angles 1. Given

2. m/R + m/S = 180 2. Definition of Supplementary Angles

3. /Q > /S 3. Given

4. m/Q = m/S 4. Definition of Congruent Angles

5. m/R + m/Q = 180 5. Substitution of Equals (4 into 2)

6. /R and /Q are supplementary angles 6. Definition of Supplementary Angles

7. QP### || RS### 7. Theorem 21 – If two lines are cut by a transversal



are parallel.


1. /BCF > /BFE 1. Given

2. /BFE > /BFC 2. Given

3. FC##$ bisects /BFG 3. Given

4. /BFC > /CFG 4. Definition of Angle Bisector

5. m/BCF = m/BFE 5. Definition of Congruent Angles

m/BFE = m/BFC

m/BFC = m/CFG

6. m/EFC + m/CFG = m/EFG 6. Postulate 7 – Protractor - Fourth Assumption –

7. /EFG is a straight angle 7. Definition of Straight Angle –

An angle whose sides are opposite rays …

8. m/EFG = 180 8. Definition of Straight Angle –

…giving a measure of 180 degrees.

9. m/EFC + m/CFG = 180 9. Substitution of Equals (6 into 8)

10. m/BCF = m/CFG 10. Transitive Property of Equality

11. m/EFC + m/BCF = 180 11. Substitution of Equals (10 into 9)

12. /EFC and /BCF are supplementary angles 12. Definition of Supplementary Angles

13. AD@#$ || EG@#$ 13. Theorem 21 – If two lines are cut by a transversal



are parallel.


1. /ABC > /CBD; /ABC > /EFH 1. Given

2. m/ABC = m/CBD 2. Definition of Congruent Angles

m/ABC = m/EFH

3. m/EFH = m/CBD 3. Substitution of Equals (2 into 2)

4. /EFH > /CBD 4. Definition of Congruent Angles

5. AD@#$ || EG@#$ 5. Corollary 20a – If two lines are cut by a transversal

so that alternate exterior angles are congruent,



1. /BCD > /D 1. Given

2. m/BCD = m/D 2. Definition of Congruent Angles

3. m/B + m/D = 180 3. Given

4. m/B + m/BCD = 180 4. Substitution of Equals (2 into 3)

5. /B and /BCD are supplementary angles 5. Definition of Supplementary Angles

6. BA##$ || CD### 6. Theorem 21 – If two lines are cut by a transversal



are parallel.


1. Either < ||y m. or < || m.. 1. Indirect Proof Assumption

Assume < || m..

2. /2 and /4 are supplementary angles 2. Theorem 17 – If two parallel lines are cut by

a transversal, then interior angles on the same

side of the transversal are supplementary.

3. /2 and /4 are not supplementary 3. Given

4. < ||y m 4. R.A.A. – Statements 2 and 3 contradict each other,

so our indirect proof assumption is false.




p. 302 – Lesson 8 — Theorem 22: If two lines are perpendicular to a third line, then the two lines are parallel.

1. a) If two lines are perpendicular to a third line, then the two lines are parallel.

b)

c) Given: < > t

m > t

d) Prove: < || m

e) Statement Reason

1. < > t ; m > t 1. Given

2. /2 is a right angle 2. Definition of a Right Angle

/6 is a right angle

3. /2 > /6 3. Theorem 11 – If you have right angles, then

those right angels are congruent.




2. Statement Reason

1. t > <1 1. Given

2. <1 || <3 2. Given

3. t > <3 3. Theorem 18 – If a given line is perpendicular to

one of two parallel lines, then it is perpendicular

to the other.

4. <2 || <3 4. Given

5. t > <2 5. Theorem 18 – If a given line is perpendicular to


to the other.

3. Statement Reason

1. DE### > BC### 1. Given

2. m /B = 90 2. Given

1 2

3 4

5 6

7 8

<

m

t

3. (continued)

Statement Reason

3. /B is a right angle 3. Definition of Right Angle

4. AB### > BC### 4. Definition of Perpendicular Lines

5. AB### || DE### 5. Theorem 22 – If two lines are perpendicular to

a third line, then the two lines are parallel.

4. Statement Reason

1. /H and /HEF are supplementary angles 1. Given

2. HG### || FE### 2. Theorem 21 – If two lines are cut by a transversal



are parallel.

3. EG### > HG### 3. Given

4. EG### > FE### 4. Theorem 18 – If a given line is perpendicular to


to the other.

5. Statement Reason

1. /BDE is a right angle 1. Given

2. BD### > DE### 2. Definition of Perpendicular Lines

3. AB### > BD### 3. Given

4. AB### || DE### 4. Theorem 22 – If two lines are perpendicular to


5. /A > /E 5. Theorem 16 – If two parallel lines are cut by

a transversal, then alternate interior angles

are congruent.

6. m/A = m/E 6. Definition of Congruent Angles

6. Statement Reason

1. /AEB and /CFD are right angles 1. Given

2. AG### > CB### 2. Definition of Perpendicular Lines

HD### > CB###

3. AG### || HD### 3. Theorem 22 – If two lines are perpendicular to


4. /ACG > /HDC 4. Postulate 11 – If two parallel lines are cut


are congruent.



7. Statement Reason

1. t > < ; < is not parallel to m 1. Given

2. Either t > m or t >y m. 2. Indirect Proof Assumption

Assume t > m.

3. < || m 3. Theorem 22 – If two lines are perpendicular to


4. This contradicts the given < ||y m. So 4. R.A.A.

t > m is false, and t >y m must be true.

8. m/A = 47

a) AB || CD Theorem 22 – If two lines are perpendicular to a third line,


b) /A > /D Theorem 16 – If two parallel lines are cut by a transversal,


9. AB || CD Theorem 22 – If two lines are perpendicular to a third line,


m/ABE + m/BEC = 180 Theorem 17 – If two parallel lines are cut by a transversal

37 + m/BEC = 180 then interior angles on the same side of the transversal are supplementary.

m/BEC = 143

/ABE > /BED Theorem 16 – If two parallel lines are cut by a transversal,

37 = m/BED then alternate interior angles are congruent.

10. m/BAC = 55 Linear Pair of angles is supplementary.

m/DCE = 125Theorem 22 – If two lines are perpendicular to a third line,


Theorem 16 Corollary – If two parallel lines are cut by a transversal,

then alternate exterior angles are congruent.

11. 4 If the line intersects the plane, then 3 points determine the plane,

plus 1 point outside the plane can be paired with one of the three points

to determine the line.

12. AB@#$ The planes intersect with the line AB being the intersection.


p. 305 – Lesson 9 — Theorem 23: If two lines are parallel to a third line, then the two lines are parallel to each other.

1. a) Theorem 23 – If two lines are parallel to a third line, then the two lines are parallel to each other.

b)

c) Given: < || r ; m || r

d) Prove: < || m

e) Statement Reason

1. < || r ; m || r 1. Given

2. Assume < ||ym 2. Indirect Proof Assumption

3. Line < must intersect line m at a point P. 3. If two coplanar lines are not parallel, then they

do intersect. (Contrapositive of Definition of

Parallel Lines)

4. Since line m passes through point P (and is 4. This is not possible by Postulate 9.Through a

parallel to line r ) and line < passes through point not on a line, there can be only one line

point P (and is parallel to line r ), we have two parallel to a given line.

lines through the same point parallel to line r .

5. < || m 5. R.A.A.

2. a || d 4. e || c 6. a || b

3. e || d 5. d || c 7. a || e

8. <1 || <3 Theorem 20 – Alternate Interior Angles Congruent

<1 || <2 Theorem 21 – Same Side Interior Angles Supplementary

<2 || <3 Theorem 23 – Two lines parallel to a third, or

Theorem 15 – Vertical Angles Congruent; Theorem 21 – Same Side Interior Angles Supplementary

t 2 || t 4 Theorem 20 Corollary – Alternate Exterior Angles Congruent

9. Yes c || b Theorem 21 – Same Side Interior Angles Supplementary

c || a Theorem 19 – Corresponding Angles Congruent

Therefore, a || b . Theorem 23

<

mP

r

•



10. no b || c Theorem 22 – Two lines perpendicular to third line are parallel to each other.

b || a Theorem 20 Corollary – Alternate Exterior Angles Congruent

Therefore, a || c. Theorem 23

11. no b || c Theorem 19 – Corresponding Angles Congruent

b || a Theorem 21 Corollary – Same Side Exterior Angles Supplementary

Therefore, a || c. Theorem 23

12. yes a || c Theorem 21 Corollary – Same Side Exterior Angles Supplementary

b || c Theorem 20 Corollary – Alternate Exterior Angles Congruent

Therefore, a || b. Theorem 23


1. /1 > /2 1. Given

2. p || q 2. Theorem 20 – If two lines are cut by a transversal

so that alternate interior angles are congruent,


3. q || r 3. Given

4. p || r 4. Theorem 23 – If two lines are parallel to a third

line, then the two lines are parallel to each other.


1. Either line t intersects line q or line t does 1. Indirect Proof Assumption

not intersect line q. Assume line t does not

intersect line q at point R .

2. t || q and t contains point R . 2. Definition of Parallel Lines – Lines which are

coplanar and do not intersect

3. p ||q ; transversal t intersects line p at point R . 3. Given

4. t || p is a contradiction 4. Theorem 23 – If two lines are parallel to a third


5. The assumption that t does not intersect 5. R.A.A.

q is false. we conclude t intersects q.


1. Draw auxiliary line ZW@##$ parallel to UV@#$ where 1. Postulate 9 – In a plane, through a point not on a

point Z is on the U side of XW@##$ a given line, there is exactly one line parallel to

the given line.

2. UV@#$ || XY@#$ 2. Given

15. (continued)

Statement Reason

3. ZW@##$ || XY@#$ 3. Theorem 23 – If two lines are parallel to a third


4. /VUW > /ZWU 4. Theorem 16 – If two parallel lines are cut by

/WXY > /XWZ a transversal, then alternate interior angles

are congruent.

5. m/VUW = m/ZWU 5. Definition of Congruent Angles

m/WXY = m/XWZ

6. m/XWU = m/XWZ + m/ZWU 6. Postulate 7 – Protractor - Fourth Assumption –

“Angle Addition” Assumption

7. m/XWU = m/WXY + m/VUW 7. Substitution of Equals (5 into 6)

8. m/XWU = m/VUW + m/WXY 8. Commutative Property of Addition

16. Theorem 23 would be true when the three lines are in space and when the three lines are in the same plane as assumed.

A simple example would be the lines determined by the gable and two edges of the roof of a small rectangular building.

Essentially, three planes are determined by the three non-coplanar lines where each pair of parallel lines determine one plane.

17. Theorem 22 Rewritten – If two planes are perpedicular to a third plane, then the two planes are parallel.

This statement is not always true. In an ordinary room, if the two planes are the floor and ceiling perpendicular to a wall, then

the statement is true. If however, the two planes are two intersecting walls perpendicular to the floor, then the statement is

not true.

Theorem 23 Rewritten – If two planes are parallel to a third plane, then the two planes are parallel to each other.

This statement is true. Think of the three planes being the floors of a three story building.


1. /1 > /3 1. Given

2. p || q 2. Theorem 19 – If two lines are cut by a transversal



3. p || r 3. Given

4. q || r 4. Theorem 23 – If two lines are parallel to a third


5. /3 and /4 are supplementary angles 5. Theorem 17 – If two parallel lines are cut by a

transversal, then interior angles on the same side

of the transversal are supplementary.




p. 307 – Lesson 10 — Theorem 24: If two parallel planes are cut by a third plane, then the two lines of intersection are parallel.

1. a) Theorem 24: If two parallel planes are cut by a third plane, then the two lines of intersection are parallel.

b)

c) Given: Plane P is parallel to plane Q.

Plane R intersects plane P and plane Q in lines m and n.

d) Prove: Line m is parallel to line n.

e) Statement Reason

1. Plane P is parallel to plane Q. 1. Given

2. Plane R intersects plane P and plane Q 2. Given

in lines m and n.

3. Lines m and n are straight lines. 3. Postulate 5 – If two planes intersect, then the

intersection is a unique line.

4. Lines m and n lie in the same plane R. 4. Step 2 – Given.

5. Line m (in plane P) and line n (in plane Q) 5. Step 1 – Plane P and plane Q are parallel, so

have no points in common. the planes have no points in common.

6. m || n 6. Definition of Parallel Lines – lines which are

coplanar and do not intersect.

2. a) Through a given point not on a given line, one and only one plane can be passed parallel to the given line.

False – not “one and only one”; infinitely many.

b) Through a given point not on a given plane, one and only one line can be drawn parallel to the given plane.

False – not “one and only one”; infinitely many.

c) If one of two perpendicular lines is parallel to a plane, the other is also parallel to the plane.

False – only one case; infinitely many others.

d) Two lines parallel to the same plane are perpendicular. False – only one case; Two lines can be in an infinite number

of positions in space, be parallel to each other, and at the same time be parallel to a given plane.

e) Two planes perpendicular to a third plane re perpendicular to each other.

False – only one case (box corner); infinitely many where the two given planes are not perpendicular.

f) Two planes parallel to the same line are perpendicular.

False – only one case; infinitely many cases where two planes are parallel.

Q

Rm

n

P

3. a) always true c) always true e) sometimes true g) always true

b) sometimes true d) always true f) sometimes true h) always true

4. a) /E – AB – L and /J – DC – L

/F – AB – L and /H – DC – L

b) /G – AB – E and /J – DC – K

/F – AB – G and /H – DC – K

c) /F – AB – G and /J – DC – L

/E – AB – G and /H – DC – L

also

/E – AB – L and /H – DC – K

/F – AB – L and /J – DC – K

5. 2) For any two different points, there is exactly one line containing them.

3) If two different planes intersect, the intersection is a unique line.

5) Definition of perpendicular lines

6) Definition of right angle

7) Addition property of equality

8) Substitution of equals ( 90 + 90 = 180)

9) Definition of supplementary angles

10) Theorem 21 – If two lines are cut by a transversal so that interior angles on the same side of the transversal are

supplementary, then the two lines are parallel.

6. 2) Postulate 5 – If two different planes intersect, the intersection is a unique line.

3) Definition of line perpendicular to a plane – If a line is perpendicular to a plane,

it is perpendicular to every line in the plane that intersects it.

7. 4) Definition of right angle

6) Definition of perpendicular planes – If two planes intersect and any line in one of them is perpendicular to their line of

intersection and to the other plane, then the two planes are perpendicular.

8. 3) Postulate 5 – If two different planes intersect, the intersection is a unique line.

7) Definition of perpendicular lines

8) Definition of right dihedral angle – the dihedral angle has plane angles that are right angles.

d) /G – AB – F and /E – AB – L

/E – AB – G and /F – AB – L

also

/H – DC – L and /J – DC – K

/H – DC – K and /J – DC – L

e) yes; yes


geometry: a complete course€¦ · lesson 3 - theorem 15 - “if two lines intersect, then the...

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