introduction to geometry€¦ · introduction to geometry . 2 building blocks of geometry i. three...
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Introduction to
Geometry
2
Building Blocks of Geometry
I. Three building blocks of geometry: points, lines, and planes.
1. A point is the most basic building block of geometry. It has no size. It only has
location. You represent a point with a dot, and you name it with a capital letter. The
point shown is called A.
Can lie on a ____________ and/or a _____________.
2. A line is a straight, continuous arrangement of infinitely many points. It has infinite length, but no thickness. It extends forever in two directions.
How many points name a line? _____________
Example:
Name the lines: ____________
3. A ________________ has length and width, but no thickness. A plane extends in 2 dimensions infinitely.
Represented by a shape like _____________________.
Example:
II. Collinear and Coplanar Points
A. Collinear Points are points that lie on the same ________________.
Example:
Which points are collinear? _____________
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B. Coplanar points are points that lie on the same __________________.
Example:
Which points are coplanar? _____________
III. Segments and Rays
A. A line segment consists of two points called the ____________ of the segment and all the
points between them that are collinear with the two points.
You can write line segment AB, using a segment symbol, as 𝐴𝐵̅̅ ̅̅ or 𝐵𝐴̅̅ ̅̅ . There are two ways to write the length of a segment. You can write AB = 2 in., meaning the distance from A to B is 2 inches. You can also use an m for “measure” in front of the segment name, and write the distance as m𝐴𝐵̅̅ ̅̅ = 2 in. If no measurement units are used for the length of a segment, it is understood that the choice of units is not important or is based on the length of the smallest square in the grid.
Examples:
The above example may be symbolized by _______ or ________.
The second example may be symbolized by _______ or ________.
B. A ray begins at a point and extends infinitely in one direction. The initial point is called the endpoint. You need two letters to name a ray. The first letter is the endpoint of the ray, and the second letter is any other point that the ray passes through.
AB = 2 in., or m𝐴𝐵̅̅ ̅̅ = 2 in.
MN = 5 units, or m𝑀𝑁̅̅ ̅̅ ̅ = 5 units
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Example:
Name the example above: ___________________
Name the example above: ___________________
IV. Intersections
A. Two or more geometric figures _________________ if they have one or more points in common.
B. The _______________________ of two or more geometric figures is the set of __________ the figures have in common.
C. Name the intersection of the two lines below: ________________
The intersection of two different lines is a _____________.
D. Name the intersection of the two planes below: ______________
The intersection of two different planes is a ______________.
V. Postulates or Axioms – rules accepted __________________________.
VI. Theorems – rules that are _______________.
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VII. Segment Addition Postulate:
If B is _______________ A and C, then _______________.
If __________________________, then B is _________________ A and C.
Example 1:
DF= _______
Example 2: Suppose M is between L and N. Use the Segment Addition Postulate to solve for the
variable. Then find the lengths of LM, MN, and LN.
1. LM = 3x + 8, MN = 2x – 5, LN = 23
2. LM w1
22
, MN w3
32
, LN w5 2
VIII. Congruent Segments – Line segments that have the same ________________.
In the diagram below, you can say “the length of AB is equal to the length of EF ,” or you can
say “ AB is congruent to EF .” The symbol means “is congruent to.”
Lengths are equal: AB = EF
Segments are congruent: AB CD
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Skills Practice:
1. Name a line, segment, and ray in this figure:
2. What is wrong with the following: 𝐴𝐵⃗⃗⃗⃗ ⃗ ≅ 𝐶𝐷⃡⃗⃗⃗ ⃗?
3. Name each of the lines in two different ways.
a. b. c.
4. Use a ruler to draw each line, ray, or segment. Don’t forget to use arrow heads to show that the
line extends indefinitely and to label the two points.
a. AB b. KL c. XY
5. Name each line segment.
a. b. c.
6. Name the ray in two different ways.
a. b. c.
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7. Draw a plane containing four coplanar points P, Q, R, and S, with exactly three collinear points Q,
R, and S.
For Exercises 8-10, use the figure located on the right.
8. Draw AB where point B has coordinates (2,-6).
9. Draw OM with endpoint (0,0) that goes through point M(2,2).
10. Draw CD through points C(-2,1) and D(-2,-3).
11. Given AC= 38. Find AB and BC.
12. Find the length of MO.
13. Given the following pairs of congruent segments, label the figure below.
AB DE , AC DF , and BC EF
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Midpoint and Distance
I. Midpoint - a point that divides, or ______________ the segment into two ______________ _______________.
Example 1: Identify the midpoint of the following line segment below.
1 2 3 4 5 6
123456789
x
y X
Y
Z
Endpoint X ___________ Endpoint Y ___________ Midpoint Z __________
Example 2: Find the midpoint of the following line segment below.
-3 -2 -1 1 2 3 4 5
1
2
3
4
5
x
y
B
A
M
What are the coordinates of the endpoints and midpoint of line segment ?AB
Endpoint A ___________ Endpoint B ___________ Midpoint M __________
Looking at the endpoints and middle coordinates, is there another way to find the middle of a
line segment?
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Midpoint Formula:
Example 1: D( -2, 6) and F(3, 4)
Example 2: The midpoint of ST is M (2, 4). One endpoint is S(-1, 7). Find the coordinates of
T. Let (x, y) be the coordinates of T.
Example 3: The midpoint of JK is M
2
1,0 . One endpoint is J (2, -2). Find the coordinates of
the other endpoint.
II. Segment bisector – a line, ray, or segment that intersects a segment at its ________________.
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Perquisite Skills- Simplifying Radicals
Simplify.
1. 90 2. 165 3. 375 4. 180
III. Distance
You can think of a coordinate plane as a grid of streets with two sets of parallel lines running perpendicular to each other. Every segment in the plane that is not in the x- or y-direction is the hypotenuse of a right triangle whose legs are in the x- and y-directions. So you can use the Pythagorean Theorem to find the distance between any two points on a coordinate plane.
In Steps 1 and 2, find the length of each segment by using the segment as the hypotenuse of a right triangle. Simply count the squares on the horizontal and vertical legs, then use the Pythagorean Theorem to find the length of the hypotenuse. Step 1 Use each segment as the hypotenuse of a right triangle. Draw the legs along the grid lines. Find the length of each segment.
Step 2 Graph each pair of points, then find the distances between them.
a. (–1, –2), (11, –7)
b. (–9, –6), (3, 10)
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What if the points are so far apart that it’s not practical to plot them? For example, what is the distance between the points A(15, 34) and B(42, 70)? A formula that uses the coordinates of the given points would be helpful. To find this formula, you first need to find the lengths of the legs in terms of the x- and y-coordinates. From your work with slope triangles, you know how to calculate horizontal and vertical distances. Step 3 Write an expression for the length of the horizontal leg using the x-coordinates. Step 4 Write a similar expression for the length of the vertical leg using the y-coordinates. Step 5 Use your expressions from Steps 3 and 4, and the Pythagorean Theorem, to find the distance between points A(15, 34) and B(42, 70). Step 6 Generalize what you have learned about the distance between two points in a coordinate plane. Distance Formula:
Recall Area Formulas:
triangleA bh1
2 rec gleA bhtan
Example 1: Find the distance of each side. Then find the area and the perimeter of the triangle.
Example 2: A map is placed on a coordinate grid. Jacksonville is located at (5, 4) and Gainesville is located
at (2, –3). How far apart are Jacksonville and Gainesville on the map? If each unit represents 10 miles,
about how far is it from Jacksonville to Gainesville?
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Skills Practice:
1. Find the coordinates of the midpoint of the segment with each pair of endpoints.
a. (12,-7) and (-6,15) b. (14, -7) and (-3,18)
2. One endpoint of a segment is (12,-8). The midpoint is (3,18). Find the coordinates of the other
endpoint.
3. In each figure below, imagine drawing the diagonals AC and BD .
a. Find the midpoint of AC and the midpoint of BD in each figure.
b. What do you notice about the midpoints for each figure.
4. Find the distance between each pair of points.
a. (10,20) and (13,16) b. (-19,-16) and (-3,14)
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5. Look at the diagram of Isabella’s and Kayleigh’s locations. Assume each block is approximately 50
meters long. What is the shortest distance, to the nearest meter, from Isabella to Kayleigh? What
is the midpoint?
6. Find the perimeter of ABC with vertices A(2,4), B(8,12), and C(24,0).
7. Find the area and the perimeter of the rectangle with vertices: A(6,8) , B(9,7), C(7,1), D(4,2)
8. Find the area and the perimeter of the square with vertices: M(-3,5), N(-1,1), O(3,3), P(1,7)
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Measure and Classify Angles
I. Angle – consists of _____________________ that have the same _____________________. The rays are the sides of the angle. The endpoint is the __________ of the angle. Example:
Denoted as _____________________ or __________________
Measured in ___________________________
Example: Name 3 different angles. _________, _________, _________
II. Measuring Angles- to measure an angle, we use a tool called a protractor.
A
C
B
K
J
N
L
M
P
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Example: Use your protractor to measure each angle. Which angle measures more than 90°?
III. Congruent angles – angles that have _______________________________________________.
IV. A ray is the angle bisector if it contains the vertex and divides the angle into two congruent
angles. In the figure at right, 𝐶𝐷⃗⃗⃗⃗ ⃗ bisects ACB so that ACD ≅ BCD.
Example: KL bisects JKM . If JKM 110 , then m JKL _____
K
J
M
L
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Example: RQ bisects PRS. What is m PRQ _______?
Example: BD bisects ABC . What is the value of x?
V. Angle Addition Postulate – If P is in the ______________________ of RST thenm RSP m PST m RST .
Example:
m ABC
m CBD
m ABD
35
15
______
3x-20
x + 40
R
P
S
Q
S
R
P
T
B
A
C
D
(5x-22)
(2x+35)
B
A
C
D
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Example:
m KLA
m BLA
m KLB
87
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______
VII. Angle Classifications
1. Acute angles - ______________________________________
2. Right angles - _______________________________________
3. Obtuse angles - ______________________________________
4. Straight angles - _____________________________________
IX. Adjacent Angles – Two angles are adjacent if they __________ a common _______________ and
__________________, but _________ common _____________________________.
Example: Adjacent Angles Example: Non-adjacent angles
L
K
B
A
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Angle Relationships
I. Vertical Angles – Two angles are vertical if their sides form two pairs of opposite ___________. The
two angles are ______________.
Def. – Opposite angles formed when 2 lines cross.
Ex.
II. Linear Pair – two ______________ angles whose noncommon sides are opposite rays.
Ex.
Example 1:
Are and2 3 a linear pair?
Are and3 4 a linear pair?
Are and1 2 vertical angles?
Are and1 3 vertical angles?
4
3
21
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III. Complementary angles – 2 angles whose sum of their measures is _________. These angles can be
________________ or nonadjacent.
Ex of complementary angles:
IV. Supplementary angles – 2 angles whose sum of their measures is __________. These angles can be
______________________ or nonadjacent.
Ex of Supplementary angles:
Example 2: Given angle A is a complement of angle B and m A 23 . What is m B ?
Example 3: Given angle M is a supplement of angle N and m M 105 . What is m N ?
Example 4: Given angle A is a complement of angle B. Find m A and m B.
m A x
m B x
(15 3)
(5 13)
Example 5: Given m P and m Q are supplementary. Find m P and m Q.
m P x
m Q x
(8 100)
(2 50)
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Skills Practice:
1. Name each angle in two different ways.
2. Draw and label each angle.
a. TAN b. BIG c. SML
3. For each figure at right, list the angles that you can name using only the vertex letter.
4. Find the measure of each angle to the nearest degree.
a. m AQB ______ c. m AQC ______ e. m XQA ______
b. m ZQY ______ d. m ZQX ______ f. m AQY ______
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5. Given the following information, mark the figure below: RA SA T H TAR HAS, ,
6. Given the following information, mark the figure below: TA GA AGT ATG, , vertical angles
7. Name the congruent angles in the figure below.
8. Ray AB is the angle bisector of angle CAD. Find the measure of angle CAD and x.
9. Solve for the missing variable below.
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10. Using the angle addition postulate and m ADC 82 , find the missing variable.
11. Classify the following angles (linear pair, acute, obtuse, right, vertical, complementary, and
supplementary).
a. c. e. and2 4
b. d. f. and2 1
12. Given that angle A and angle B are supplementary angles and the measure of angle B is 35°, what
is the measure of angle A?
13. Given that angle C and angle D are complementary angles and the measure of angle D is 35°,
what is the measure of angle C?
14. Find the missing variables.
a. b.
(2x +21)°
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Polygons
In geometry, a figure that lies in a plane is called a plane figure. A ___________ is a closed plane figure
with the following properties:
1. It is formed by three of more line segments called _____________.
2. Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common
endpoint are collinear.
Each endpoint of a side is a ____________ of the polygon. The plural of vertex is vertices.
A polygon can be named by listing the vertices in consecutive order. For
example, ABCDE and EDEAB are both correct names for the polygon at the
right.
A ______________ of a polygon is a line segment that connects two nonconsecutive vertices.
A polygon is ___________ if no line that contains a side of the polygon contains a point in the interior of
the polygon, or all diagonals lie inside the polygon.
A polygon that is not convex is called __________.
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A polygon is named by the number of its sides.
The term n-gon, where n is the number of polygon’s sides, can also be used to name a polygon. For
example, a polygon with 16 sides is a 16-gon.
To name a polygon, be sure to include the name and the vertices in consecutive order.
Examples: Name the polygons below.
Name: _____________ or ___________ Name _________________
Two polygons are congruent if and only if they are the same _______ and __________. This means all the corresponding sides are congruent and all the corresponding angles are congruent.
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For example, if quadrilateral CAMP is congruent to quadrilateral SITE, then their four pairs of corresponding angles and four pairs of corresponding sides are also congruent. When you write a statement of congruence, always write the letters of the corresponding vertices in an order that shows the correspondences.
Example: Which polygon is congruent to ABCDE? ABCDE ≅ ?
The ______________ of a polygon equals the sum of the lengths of its sides. Perimeter measures the length of the boundary of a two-dimensional figure. Example: Find the perimeter of the quadrilateral below.
Perimeter: __________cm
In an ______________________________ polygon, all sides are congruent.
In an ____________________________ polygon, all angles in the interior of the polygon are congruent.
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A _________________ polygon is a convex polygon that is both equilateral and equiangular.
Guided Practice:
Example 1: If the perimeter of the quadrilateral is 20 cm, find x.
x = _________
Example 2: What is wrong with this picture of equilateral pentagon ABCDE? Hint: there are two things
wrong with this picture.
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Skills Practice:
For Exercises 1-3, draw an example of each polygon.
1. Quadrilateral 2. Dodecagon 3. Octagon
For Exercises 4-7, classify each polygon. Assume all sides are straight.
For exercises 8-10, give one possible name for each polygon.
For exercises 11 and 12, use the information given to name the triangle that is congruent to the first
one.
11. 12.
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13. Indicate if the following figures are polygons.
14. Indicate if the following polygons are concave or convex.
15. In the figure at right, THINK POWER.
a. Find the measures a, b, and c.
b. If m P 87 and m W 165 , which
angles in THINK do you know? Write their measures.
16. Each side of a regular dodecagon measures 7 in. Find the perimeter.
17. The perimeter of an equilateral octagon is 42 cm. Find the length of each side.
18. The perimeter of ABCDE is 94 m. Find the lengths of segments AB and CD.
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Algebraic Proofs
Warm Up: Solve for x.
1. x3 5 17
2. x x4 5 8 3
3. x2( 5) 20 0
4. x 8
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Proofs
Justifications are used for EVERY step! Here is a list of justifications that CAN be used. There are other justifications that you learned and can use
as well.
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Example 1:
Complete a two column proof to prove the following:
Given: x 5
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Prove: x 7
Example 2:
Complete a two column proof to prove the following:
Given: x x6 3 3( 1) Prove: x 2
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Example 3:
Complete a two column proof to prove the following:
Given: m MRK x m KRW x m MRW3 , 6, 90 Prove: m MRK 63
Example 4:
Complete a two column proof to prove the following:
Given: AB x BC x AC5 1, 3 4, 13 Prove: BC 2
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Guided Practice:
Part A- Identify the property that justifies each statement.
1. AB = AB __________________
2. If 1 2m m and 2 4,m m then 1 4m m __________________
3. If x = y, then y = x. __________________
4. If ST = YZ, and YZ = PR, then ST = PR __________________
5. If KL = PR, then KL-AB = PR – AB __________________
6. 412 = 412 __________________
7. If b = a and b = 0, then a = 0 __________________
8. Figure A = Figure A __________________
9. If m DEF m ABC , then m DEF m GHI m ABC m GHI _________________
10. If x = y, then 3 3
x y __________________
11. If AB = CD, then CD = AB __________________
12. If 72
x , then x = 14 __________________
13. If x = 5 and b = 5, then x = b __________________
14. If XY – AB = WZ – AB, then XY = WZ __________________
15. If m A m B , and m B m C , then m A m C __________________
Part B- Use the property to complete the statement.
16. Reflexive Property: _______ = SE
17. Symmetric Property: If ________ = _________, then m RST m JKL
18. Transitive Property: m F m J and ________ = _________, then m F m L
19. Addition Property: If RS = TU, then RS +20 = _____________
20. Multiplication Property: If 1 2m m then 3 1( )m __________
21. Substitution Property: If a = 20, then 5a = __________
Part C- Complete the two-column proofs below using the appropriate properties.
22. Given:8 34 6x 23. Given:5 3 4 2( ) ( )x x
Prove: 5x Prove: 23x
Statement Reason
Statement Reason
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24. Given: 4 7 6 7x x 25. Given:1
97
x y
Prove: 7x Prove: 7 63y x
Statement Reason
26. Given: 7 11 4 19x x 27. Given:14 3 19 23x x
Prove: 10x Prove: 4x
28. Given: 4 2 11 76( )x 29. Given:14 1 7 4( ) ( )x x
Prove: 4x Prove: 2x
30. Given: ,SU LR TU LN Prove: ST NR
Statements Reasons
1. 𝑆𝑈̅̅̅̅ ≅ 𝐿𝑅̅̅̅̅ , 𝑇𝑈̅̅ ̅̅ ≅ 𝐿𝑁̅̅ ̅̅ 2. _____________________________
1. Given 2. Definition of Congruent Segments
3. _______________ ; ______________
3. Segment Addition Postulate
4. ________________________________
4. Substitution Property
5. ______________________________
5. Substitution Property
6. _____________________________ 7. ______________________________
6. ____________________________ 7. _____________________________
Statement Reason
S U
L R
T
N
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Skills Practice:
Identify the property that justifies each statement.
1. If HJ + 5 = 20, then HJ = 15 ____________________
2. If XY + 20 = YW and XY + 20 = DT, then YW = DT ____________________
3. If 1 2 90m m and 2 3m m , then 1 3 90m m ____________________
4. If 1 1
2 2AB EF , then AB = EF ____________________
5. If 3
2 52
( )x , then 2 3 5x ____________________
6. If 4 35m and 5 35,m then 4 5m m ____________________
7. If 2 2
3 3AB CD , then 2 2AB CD ____________________
8. If EF = GH and GH = JK, then EF = JK _____________________
Use the property to complete the statement.
9. Reflexive Property: If AB AB then ____________
10. Symmetric Property: If AB = CD, then CD = ___________
11. Transitive Property: If m E m F and m F m G , then _____________
12. Multiplication Property: If RS = TU, then x(RS) = ____________
13. Division Property: If 3 1 3 2( ) ( )m m , then 1m _______________
14. Transitive Property: If a = bc and bc = de, then _______________
15. Substitution Property: If x = 3c and r = 5x + 7, then _____________
Create two-column proofs that prove the following statements using properties.
16. Given: 19 2 9x 17. Given: 3 2 22x
Prove: 5x Prove: 8x
Statements Reasons
17. Given: 109 3 5 5 4n( )
Prove: 8n
Statements Reasons
Statements Reasons
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18. Given: 1 and 2 are supplementary, m x m1 4 , 2 80
Prove: 1 100m
Statements Reasons
19. Given: 1 and 2 are vertical angles, m x m x1 2 21, 2 4
Prove: x 10.5
Statements Reasons
21
2
1
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Prove Statements about Segments and Angles
Writing a two-column proof is a formal way of organizing your reasons to show a statement is true.
Each reason in the right-hand column is the explanation for the corresponding statement.
Write a two-column proof for the situations below.
Example 1: Given: m m1 3
Prove: m EBA m DBC
Statements Reasons
m EBA m m3 2
m EBA m m1 2
m m m DBC1 2
Transitive Property of Equality
Example 2: Given: AC = AB + AB
Prove: AB = BC
Statements Reasons
AB + BC = AC
AB + AB = AB + BC
AB = BC
The reasons used in a proof can include definitions, properties, postulates, and theorems. A
______________ is a statement that can be proven. Once you have proven a theorem, you can use the
theorem as a reason in other proofs.
Theorems:
Congruence of Segments
Segment congruence is reflexive, symmetric, and transitive.
o Reflexive – For any segment AB, AB AB
o Symmetric – If AB CD, then CD AB
o Transitive – If AB CD and CD EF , then AB EF
Congruence of Angles
Angle congruence is reflexive, symmetric, and transitive.
o Reflexive – For any angle A, A A
o Symmetric – If A B then B A
o Transitive – If A B and B C, then A C
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Example 3: Prove this property of midpoints. If you know that M is the midpoint of AB, prove
that AB is two times AM and AM is one half of AB.
Given: M is the midpoint of AB
Prove: AB = 2 ● AM and AM = AB1
2
Statements Reasons
M is the midpoint of AB Given
Definition of Midpoint
AM = MB Definition of congruent segments
Segment Addition Postulate
AM + AM = AB
2 ● AM = AB
AM = AB1
2
Example 4: Complete the proof below.
Given: SU LR TU LN,
Prove: ST NR
Statements Reasons
SU LR TU LN,
Definition of Congruent Segments
SU = ST + TU
LR = LN + NR
ST + TU = LN + NR
ST + LN = LN + NR
ST = NR
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Skills Practice
Name the property illustrated by the statement.
1. If DG CT , then CT DG Property: __________________________
2. VWX VWX Property: __________________________
3. If JK MN and MN XY , then JK XY Property: __________________________
4. YZ = ZY Property: __________________________
Use the property complete the statement.
5. Reflexive Property of Congruence: _______ SE
6. Symmetric Property of Congruence: If ______ _______, then RST JKL
7. Transitive Property of Congruence: If F J and ______ _______, then F L
Complete the proofs below.
8. Given: AB = 5, BC = 6
Prove: AC = 11
Statements Reasons
AB = 5, BC = 6
Segment Addition Postulate
Substitution Property
9. Given: RT = 5, RS = 5, RT TS
Prove: RS TS
Statements Reasons
Transitive Property of Equality
RT = TS
RS = TS
T
R
S
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10. Given: m 1 45 and m 2 45
Prove: AB is the bisector of DAC
Statements Reasons
m 1 45 and m 2 45
Substitution Property of Equality
1 2
11. Given: FD bisects EFC and FC bisects DFB
Prove: EFD CFB
12. Given: and1 2 are complementary, 1 3 , and 2 4
Prove: and3 4 are complementary
Statements Reasons
Given
1 3
Given
Def. of Congruence
Def. of Congruence
m m1 2 90
Substitution Property
Def. of Complementary Angles
Statements Reasons
Given
Given
EFD DFC
DFC CFB
Transitive Property of Congruence
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13. Given: and1 2 form a linear pair and m m2 2( 1)
Prove: m 1 60
Statements Reasons
Given
Given
and1 2 are supplementary angles
Def. of Supplementary angles
Substitution Property
Division Property of Equality
14. Given: and1 2 are complementary and m 2 46
Prove: m 1 44
Statements Reasons
Given
Given
m m1 2 90
15. Given: m m1 2 180 and m 1 62
Prove: m 2 118
Statements Reasons
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Prove Angle Pair Relationships
When two lines intersect, pairs of vertical angles and linear pairs are formed.
Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.
1 and 2 form a linear pair, so 1 and 2 are supplementary and m m1 2 180
Vertical Angles Congruence Theorem: Vertical angles are congruent.
1 3, 2 4
Prove the Vertical Angles of Congruence Theorem-
Given: 4 and 3 are vertical angles
Prove: 4 3
Statements Reasons
2 and 4 are a linear pair
Def. of Linear Pair
2 and 4 are supplementary
Linear Pair Postulate
Def. of Supplementary
m m2 3 180
Substitution Property
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Examples:
Use the diagram below to answer the following questions. Note that the diagram is not drawn to scale.
1. If m 1 112 , find m m and m2, 3, 4.
2. If m 2 67 , find m m and m1, 3, 4.
3. If m 4 71 , find m m and m1, 2, 3.
4. Multiple Choice: Which equation can be used to find x?
A. x32 (3 1) 90
B. x32 (3 1) 180
C. x32 3 1
D. x3 1 212
5. Solve for x in Example 4 above.
6. Find m TPS in Example 4 above.
Guided Practice
1. Describe the relationship between the angle measures of complementary angles, supplementary
angles, vertical angles, and linear pairs.
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2. Identify the pair(s) of congruent angles in the figures below. Explain how you know they are
congruent.
a. c.
b. d.
3. Find the measure of each numbered angle.
a. m 2 57 c. m 5 22 e. m 1 38
b. m x13 4 11, d. 9 and 10 are f. m x2 4 26
m x14 3 1 complementary. m x3 3 4
m7 9, 41
44
Skills Practice
Find the value of x in each figure.
1. 2. 3.
4. 5. 6.
7. What is the value of x if PQR and SQT are vertical angles and m PQR 47 and
m SQT x3 2?
8. Find the measure of an angle that is supplementary to B if the measure of B is 58°
45
Find the measure of each numbered angle and name the theorems that justify your work.
9. m x1 10 10. m x4 2 5 11. m x6 7 24
m x2 3 18 m x5 4 13 m x5 14
12. x = _______
m MAT ________
13. x = _________
m PIR ________
m RIM ______
12. Write a two-column proof.
Given: 1 and 2 form a linear pair and 2 and 3 are supplementary
Prove: 1 3