1. Lesson 1.5 : Describing Angle Pair Relationships

2. Adjacent Angles
Nonadjacent Angles
Complementary Angles
Supplementary Angles
Linear Pair
Vertical Angles

3. EXAMPLE 1
Identify complements and supplements
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.
SOLUTION
Because 32°+ 58° = 90°, BAC and RST are complementary angles.
Because 122° + 58° = 180°, CAD and RST are supplementary angles.
Because BAC and CAD share a common vertex and side, they are adjacent.

4. GUIDED PRACTICE
for Example 1
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1.
FGK and GKL,
HGK and GKL,
FGK and HGK
ANSWER

5. GUIDED PRACTICE
for Example 1
Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent angles? Explain. 2.
No, they do not share a common vertex.
No, they have common interior points.
ANSWER

6. EXAMPLE 2
Find measures of a complement and a supplement
Given that 1 is a complement of and m = 68°,
find m
SOLUTION
a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.
m = 90° – m = 90° – 68° = 22°

7. EXAMPLE 2
Find measures of a complement and a supplement
b. Given that is a supplement of 4 and m = 56°,
find m 3.
SOLUTION
b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship.
m 3 = 180° – m 4 = 180° –56° = 124°

8. EXAMPLE 3
Find angle measures
Sports
When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.

9. EXAMPLE 3 Find angle measures SOLUTION STEP 1
Use the fact that the sum of the measures of supplementary angles is 180°.
m BCE + m ∠ ECD = 180°
Write equation.
(4x + 8)° + (x + 2)° = 180°
Substitute.
5x + 10 = 180
Combine like terms.
5x = 170
Subtract 10 from each side.
x = 34
Divide each side by 5.

10. EXAMPLE 3
Find angle measures
SOLUTION
STEP 2
Evaluate: the original expressions when x = 34.
m BCE = (4x + 8)° = ( )° = 144°
m ECD = (x + 2)° = ( )° = 36°
The angle measures are 144° and 36°.
ANSWER

11. GUIDED PRACTICE
for Examples 2 and 3
3. Given that is a complement of and m = 8o, find m 1.
82o
ANSWER
4. Given that is a supplement of and m = 117o, find m
63o
ANSWER
LMN and PQR are complementary angles. Find the measures of the angles if m LMN = (4x – 2)o and m PQR = (9x + 1)o.
ANSWER
26o, 64o

12. EXAMPLE 4
Identify angle pairs
Identify all of the linear pairs and all of the vertical angles in the figure at the right.
SOLUTION
To find vertical angles, look or angles formed by intersecting lines.
1 and are vertical angles.
ANSWER
To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.
1 and 4 are a linear pair and are also a linear pair.
ANSWER

13. EXAMPLE 5
Find angle measures in a linear pair
Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.
ALGEBRA
SOLUTION
Let x° be the measure of one angle. The measure of the other angle is 5x°. Then use the fact that the angles of a linear pair are supplementary to write an equation.

14. Find angle measures in a linear pair
EXAMPLE 5
Find angle measures in a linear pair
xo + 5xo = 180o
Write an equation.
6x = 180
Combine like terms.
x = 30o
Divide each side by 6.
The measures of the angles are 30o and 5(30)o = 150o.
ANSWER

15. GUIDED PRACTICE
For Examples 4 and 5
Do any of the numbered angles in the diagram below form a linear pair? Which angles are vertical angles? Explain. 6.
ANSWER
No, no adjacent angles have their noncommon sides as opposite rays, and , and 5, and 6, these pairs of angles have sides that from two pairs of opposite rays.

16. GUIDED PRACTICE
For Examples 4 and 5
7. The measure of an angle is twice the measure of its complement. Find the measure of each angle.
ANSWER
60°, 30°

17. 1. 1 and 2 are supplementary. If m 1 = 97 , find m 2.
Daily Homework Quiz
and are supplementary. If m = 97 , find m o
ANSWER 83 o
and are complementary angles. If m 3= 74, Find m o
ANSWER 16 o

18. Daily Homework Quiz 3. Find m EFH. 96 ANSWER 4. Find m ABC. ANSWER 36

19. Daily Homework Quiz
5. Is it possible to draw a figure that contains exactly one pair of vertical angles? Explain.
No; once you have drawn a pair of vertical angles, you have drawn two pairs of opposite rays. This automatically gives another pair of vertical angles.
ANSWER

20. Lesson 1.6 :
Classifying Polygons

21. EXAMPLE 1
Identify polygons
Tell whether the figure is a polygon and whether it is convex or concave. a. b. c. d.
SOLUTION
Some segments intersect more than two segments, so it is not a polygon. a. b.
The figure is a convex polygon.
Part of the figure is not a segment, so it is not a polygon. c. d.
The figure is a concave polygon.

22. EXAMPLE 2
Classify polygons
Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. a. b.
SOLUTION
The polygon has 6 sides. It is equilateral and equiangular, so it is a regular hexagon. a.
The polygon has 4 sides, so it is a quadrilateral. It is not equilateral or equiangular, so it is not regular. b.

23. EXAMPLE 2
Classify polygons
Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. c.
SOLUTION c.
The polygon has 12 sides, so it is a dodecagon. The sides are congruent, so it is equilateral. The polygon is not convex, so it is not regular.

24. GUIDED PRACTICE
for Examples 1 and 2
Sketch an example of a convex heptagon and an example of a concave heptagon. 1.
ANSWER

25. GUIDED PRACTICE
for Examples 1 and 2
Classify the polygon shown at the right by the number of sides. Explain how you know that the sides of the polygon are congruent and that the angles of the polygon are congruent. 2.
Quadrilateral. They all have the same measure; they are all right angles.
ANSWER

26. EXAMPLE 3 Find side lengths
A table is shaped like a regular hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.
ALGEBRA
SOLUTION
First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent.
3x + 6
4x – 2 =
Write equation. 6 =
x – 2
Subtract 3x from each side. 8 = x
Add 2 to each side.

27. EXAMPLE 3
Find side lengths
Then find a side length. Evaluate one of the expressions when x = 8. 30
3(8) + 6 =
3x + 6
The length of a side of the table is 30 inches.
ANSWER

28. GUIDED PRACTICE
for Example 3
The expressions 8y° and ( 9y – 15 )° represent the measures of two of the angles in the table in Example 3. Find the measure of an angle. 3.
120o
ANSWER

29. Daily Homework Quiz
1. Draw a convex hexagon.
ANSWER
2. This figure shows the tiles on a kitchen floor. What type of polygon are the tiles? Are they regular polygons?
quadrilaterals ; not regular
ANSWER

30. Daily Homework Quiz
3. This figure is a regular polygon. Find the length of each side.
ANSWER 2