Geometry 1 6 Angle Pair Relationships Essential Question

Geometry 1. 6 Angle Pair Relationships Essential Question: How can you use angle pairs to find angle measures? mbhaub@mpsaz. org

Topics/Objectives • Know what Vertical Angles are. • Identify Linear Pairs. • Solve problems with Complementary Angles. • Solve problems with Supplementary Angles. • To define adjacent angles. 11 March 2021 Geometry 1. 6 Angle Pair Relationships 2

Another way to name angles… 1 Sometimes, for clarity and convenience, we will use a single number inside the angle to name it. This is 1. 11 March 2021 Geometry 1. 6 Angle Pair Relationships 3

More than one angle 1 11 March 2021 2 3 Geometry 1. 6 Angle Pair Relationships 4

Vertical Angles 1 3 4 2 Two angles are vertical angles if their sides form two pairs of opposite rays. 1 & 2 are vertical angles. 3 & 4 are vertical angles. 11 March 2021 Geometry 1. 6 Angle Pair Relationships 5

Vertical Angles Property Vertical Angles are congruent. 60° 11 March 2021 ? 60° Geometry 1. 6 Angle Pair Relationships 6

Linear Pair Two adjacent angles Common Side are a linear pair if their noncommon sides are opposite 1 & 2 are a linear pair. rays. 2 1 Noncommon sides 11 March 2021 Geometry 1. 6 Angle Pair Relationships 7

Linear Pair Property The sum of the angles of a linear pair is 180°. 70° 11 March 2021 110° ? Geometry 1. 6 Angle Pair Relationships 8

Example 1 2 1 5 3 4 a. Are 1 and 2 a linear pair? Yes b. Are 4 and 5 a linear pair? no c. Are 3 and 5 vertical angles? no d. Are 1 and 3 vertical angles? Yes 11 March 2021 Geometry 1. 6 Angle Pair Relationships 9

Example 2 These angles form a Find the measure of the three angles. linear pair. The sum is 180°. 130° 2 50° These are vertical angles, and congruent. 3 1 50° 130° These angles are vertical angles. Vertical angles are congruent. 11 March 2021 Geometry 1. 6 Angle Pair Relationships 10

Example 3 A B (4 x + 30)° E (6 x – 10)° page 1 Solve for x, then find the measure of each angle. C D AEB and BEC form a linear pair. What do we know about the sum of the angles of a linear pair? The sum is 180°. 11 March 2021 Geometry 1. 6 Angle Pair Relationships 11

Example 3 A 94° B (4 x + 30)° E (6 x – 10)° 86° 94° D C page 2 Linear pair AEB and BEC means: (4 x + 30) + (6 x – 10) = 180 10 x + 20 = 180 10 x = 160 x = 16 Then m AEB = 4(16) + 30 = 94 and m BEC = 6(16) – 10 = 86 11 March 2021 Geometry 1. 6 Angle Pair Relationships 12

Your Turn C 145° 3 (5 x + 30)° (2 x – 4)° A B 1 2 1. Find the measure of 1, 2, 3. 11 March 2021 2. Find the measure of ABC. Geometry 1. 6 Angle Pair Relationships 13

Solutions 180° C 145° 1 35° 3 2 145° (5 x + 30)° (2 x – 4)° A B 5 x + 30 + 2 x – 4 = 180 7 x + 26 = 180 7 x = 154 x = 22 m ABC = 5(22) + 30 = 140° 11 March 2021 Geometry 1. 6 Angle Pair Relationships 14

Complementary Angles Two angles are complementary if their sum is 90°. These angles are complementary and adjacent. 65° 25° 11 March 2021 Geometry 1. 6 Angle Pair Relationships 15

Adjacent Angles A B O Adjacent angles have the same vertex, O, and one side in common, OB. They share no interior points. C There are THREE angles: You cannot use the AOB or BOA label O, since it would be unclear BOC or COB which angle that is. AOC or COA 11 March 2021 Geometry 1. 4 Angles and Their Measure 16

RST and VST are NOT adjacent angles. R V S 11 March 2021 T Geometry 1. 4 Angles and Their Measure 17

Complementary Angles Two angles are complementary if their sum is 90°. These angles are complementary and 30° 60° 11 March 2021 nonadjacent. Geometry 1. 6 Angle Pair Relationships 18

Supplementary Angles are supplementary if their sum is 180°. The angles are adjacent and supplementary (and a linear pair). 70° 11 March 2021 110° Geometry 1. 6 Angle Pair Relationships 19

Supplementary Angles are supplementary if their sum is 180°. The angles are nonadjacent and supplementary. 80° 11 March 2021 100° Geometry 1. 6 Angle Pair Relationships 20

Example 5(40) – 50 = 150° (5 y – 50)° Solve for y, then find m 1. 30° 1 (4 y – 10)° 150° Vertical angles are congruent, so: 5 y – 50 = 4 y – 10 y =40 11 March 2021 1 forms a linear pair with either of the 150° angles, so 1 is 30°. Geometry 1. 6 Angle Pair Relationships 21

Example Find the measure of each angle. 4 x + 5 + 3 x + 8 = 90 49° (4 x + 5)° 7 x + 13 = 90 41° (3 x + 8)° This is a right angle, the angles are complementary. Their sum is 90°. 11 March 2021 7 x = 77 x = 11 4(11) + 5 = 49° 3(11) + 8 = 41° Geometry 1. 6 Angle Pair Relationships 22

Example 4 Find the value of each 50° (3 x + 8)° (5 x – 20)° variable and the measure 50° of each labeled angle. (5 x + 4 y)° 130° 3 x + 8 = 5 x – 20 130° 5 x + 4 y = 130 -2 x = -28 5(14) + 4 y = 130 x = 14 70 + 4 y = 130 3(14) + 8 = 50° 4 y = 60 y = 15 11 March 2021 Geometry 1. 6 Angle Pair Relationships 23

Solve for x: (4 x + 40) 11 March 2021 (6 x + 10) Geometry 1. 6 Angle Pair Relationships 24

Solve for x: (12 x – 12) 11 March 2021 (5 x + 5) Geometry 1. 6 Angle Pair Relationships 25

(x + 8) Solve for x: (7 x + 2) 11 March 2021 Geometry 1. 6 Angle Pair Relationships 26

Summarize 11 March 2021 Geometry 1. 6 Angle Pair Relationships 27