Parallelism Triangles Quadrilaterals

Key TermsKey Terms

skew lines: non-coplanar lines that do not intersect (lines AB and EF are skew lines)

parallel lines: non-intersecting coplanar lines (lines AD and EF are parallel lines)

A

B C

D

E F

transversal: line that intersects two coplanar lines (line AB is a transversal)

alternate interior angles: angles that lie inside two lines and on opposite sides of the transversal (r and s are alternate interior angles)

corresponding angles: if r and s are alternate interior angles and q is a vertical angle to r, then q and s are corresponding angles.

qr

sA

B

TheoremsTheorems

AIP Theorem

If you are given two lines intersected by a transversal, and a pair of alternate interior angles are congruent, then the lines are parallel.

Restatement: Given line AB and line CD cut by transversal EF. If x y, then line AB is parallel to line CD.

x

y

A B

C D

E

F

Given: line segment GH and line segment JK bisect each other at F.

Prove: line segment GK and line segment JH are parallel.

1. GH and JK bisect at F

2. GF = FH and JF = FK

3. JFH KFG

4. ∆JFH ∆KFG

5. HJF GKF

6. JH is parallel to GK

G

HJ

K

F1. Given

2. Def. of bisector

3. VAT

4. SAS

5. CPCTC

6. AIP

Statements Reasons

PCA Corollary

Corresponding angles are congruent if you are given two parallel lines cut by a transversal.

Restatement: v w if line AB and line CD are parallel and are cut by transversal EF.

A B

C D

E

F

v

w

Given: the figure with CDE A and line LF line AB.

Prove: line LF line DE.

1. CDE A, LF AB

2. DE is parallel to AB

3. GFA LGD

4. GFA is a R.A.

5. m GFA = 90˚

6. m LGD = 90˚

7. LGD = R.A.

8. LF DE

1. Given

2. CAP

3. PCA

4. Def. of perp.

5. Def. of R.A.

6. Def. of congruence

7. Def. of R.A.

8. Def. of perp.

Statements Reasons

A B

C

D EG

L

H

Key TermsKey Terms

concurrent lines: two or more lines that all share a common point (lines AB, CD, and EF are concurrent.)

point of concurrency: the common point shared by concurrent lines. (point G is the point of concurrency.)

A

BC

DE

F

G

TheoremsTheorems

Theorem 9-13

The measures of all the angles in a triangle add up to 180.

Restatement: Given ABC. mA + m B +m C = 180.

A

B

C

Given: ABC, BA AC and mB = 65.

Prove: mC = 155.

Statements

1. BA AC, m B = 65

2. A is a R.A.

3. mA = 90

4. mA + mB +mC = 180

5. 90 + 65 + mC = 180

6. mC = 155

Reasons

1. Given

2. Def. of perp.

3. Def. of R.A.

4. ms in = 180

5. Sub.

6. SPE

A

B

C

65˚

Theorem 9-28

If one side of a right triangle is half the length of the hypotenuse, then the measure of the opposite angle is 30.

Restatement: Given right triangle ABC. If AB = 1/2BC, then mBCA is 30.

A

B

C

Given: DEF is a right triangle. D = 90 and DE = 1/2EF.

Prove: mE = 60

Statements

1. DEF is a R.T. DE = 1/2EF

2. mF = 30

3. mD = 90

4. mD + mE + mF = 180

5. 90 + mE + 30 = 180

6. mE = 60

Reasons

1. Given

2. opp. side 1/2 as long as hyp. = 30

3. Def. of R.T.

4. s add up to 180

5. Sub.

6. SPE

D

E

F

Key TermsKey Terms

diagonal: a line segment connecting two nonconsecutive angles in a quadrilateral. (segment AC is a diagonal)

parallelogram: quadrilateral that has opposite parallel lines. (ABCD is a parallelogram)�

trapezoid: quadrilateral that has one pair of opposite parallel lines and one pair of nonparallel lines. (JKLM is a trapezoid.)�

A

B C

D J

K L

M

rhombus: parallelogram that has 4 congruent sides. (QRST � is a rhombus)

rectangle: parallelogram with 4 right angles. (ABCD is a � rectangle)

square: rectangle that has 4 congruent sides. (FGHJ is a �square)

Q

R

S

T

A

B C

D

F

GH

J

TheoremsTheorems

Theorem 9-21

If the diagonals in a quadrilateral bisect each other, that quadrilateral is a parallelogram.

Restatement: Given ABCD. If AC and BD bisect each other �at E, then ABCD is a parallelogram.�

A

B

D

C

E

Given: WXYZ WT = TY and XT = TZ�

Prove: WXYZ is a parallelogram.�

Statements

1. WT = TY, XT = TZ

2. WY and XZ bisect each other.

3. WXYZ is a �parallelogram.

Reasons

1. Given

2. Def. of bisectors

3. If diagonals bisect each other, = parallelogram

W

X Y

Z

T

Theorem 9-24

A rhombus’ diagonals are perpendicular to each other.

Restatement: Given ABCD is a rhombus, then AC � BD.

A

B

C

D

Given: FGHK is a rhombus.�

Prove: GJF GJH

Statements

1. FGHK is a rhombus�

2. GK FH

3. GJK and GJH are R.A.

4. GJK GJH

Reasons

1. Given

2. Diagonals are in rhombus

3. Def. of perp.

4. R.A.

F

G

H

K

J