Geometry. 1.6 Describing Pairs of Angles
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1 Geometry
2 1.6 Day 1 Warm-up Solve. 1. 4x 0 = = 11c = 19x = 5n + 5 4n 5. 3x = 2x 5 6. x x + 17 = x 2
3 Essential Question What angle relationships occur when two lines intersect? 1.4 Perimeter and Area in the Coordinate Plane
4 What You Will Learn Identify complementary and supplementary angles. Identify linear pairs, vertical angles, and adjacent angles. Solve problems with angle relationship properties.
5 Adjacent Angles O A B C You cannot use the label O, since it would be unclear which angle that is. Adjacent angles have the same vertex, O, and one side in common, OB. They share no interior points. There are THREE angles: AOB or BOA BOC or COB AOC or COA
6 RST and VST are NOT adjacent angles. R Why not? They overlap. V S T
7 Linear Pair Two adjacent angles are a linear pair if their noncommon sides are opposite rays. Common Side & 2 are a linear pair. Noncommon sides
8 Linear Pair Property The sum of the angles of a linear pair is ?110
9 Complementary Angles Two angles are complementary if their sum is These angles are complementary and adjacent.
10 Complementary Angles Two angles are complementary if their sum is These angles are complementary and nonadjacent.
11 Supplementary Angles Angles are supplementary if their sum is These angles are supplementary and adjacent and a linear pair.
12 Supplementary Angles Angles are supplementary if their sum is The angles are supplementary and nonadjacent.
13 Example 1 In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.
14 Example 2
15 Vertical Angles Two angles are vertical angles if their sides form two pairs of opposite rays. 1 & 2 are vertical angles. 3 & 4 are vertical angles.
16 Vertical Angles Property Vertical Angles are congruent. 60? 60
17 Example 3 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
18 Example 4 Find the measure of the three angles. These are vertical angles, and congruent These angles are vertical angles. Vertical angles are congruent. These angles form a linear pair. The sum is 180.
19 Example 5 A B (4x + 30) E (6x 10) Solve for x, then find the measure of each angle. D C 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.
20 Example 5 A D 94 B (4x + 30) E (6x 10) C Linear pair AEB and BEC means: Then AEB = 4(16) + 30 = 94 and BEC = 6(16) 10 = 86 (4x + 30) + (6x 10) = x + 20 = x = 160 x = 16
21 Your Turn Work through these two problems. C A (5x + 30) B (2x 4) 1. Find the measure of 1, 2, Find the measure of ABC.
22 Your Turn Solutions 180 C A (5x + 30) B 5x x 4 = 180 7x + 26 = 180 7x = 154 x = 22 m ABC = 5(22) + 30 = 140 (2x 4)
23 Example 6 Solve for x, then find the angle measures. 6(15) = 90 A 6x (3x + 45) 3(15) + 45 = 90 B E D C Solution: AEB and DEA are a linear pair. The sum of the angles in a linear pair is x + (3x + 45) = 180 9x = 135 x = 15
24 Example 7 Solve for y, then find m 1. 5(40) 50 = 150 (5y 50) 30 1 Vertical angles are congruent, so: 5y 50 = 4y 10 y = 40 (4y 10) forms a linear pair with either of the 150 angles, so 1 is 30.
25 Example 8 Find the measure of each angle. 49 (4x + 5) 41 (3x + 8) This is a right angle, the angles are complementary. Their sum is 90. 4x x + 8 = 90 7x + 13 = 90 4(11) + 5 = 49 3(11) + 8 = 41 7x = 77 x = 11
26 Example 9 Find the value of each variable and the measure of each labeled angle (3x + 8) (5x 20) 50 3x + 8 =5x 20-2x = -28 x = 14 3(14) + 8 = 50
27
28 1. Solve for x. (4x + 40) (6x + 10) 6x 10 4x 40 2x 30 x 15
29 2. Solve for x. (12x 12) (5x + 5) (12x 12) (5x 5) x x 187 x 11
30 3. Solve for x. (7x + 2) ( x 8) (7x 2) 90 8x x 80 x 10
31 4. Solve for x & y. (7x + 4) (9y + 3) (13x + 16) (5y 5) (7x 4) (13x 16) x x 160 x 8 (9y 3) (5y 5) y y 182 y 13
32 5. Solve for x. A is supplementary to B. m A = (2x + 10) m B = (3x 5) 2x x 5 = 180 5x + 5 = 180 5x = 175 x = 35
33 Quick Review What do you know about supplementary angles? Two angles are supplementary if their sum is 180. What do you know about complementary angles? Two angles are complementary if their sum is 90.
34 Quick Review Which angles are Vertical Angles and what do you know about them? 1 2 & 3 4
35 Quick Review Which angles are linear with 4 and what do we know about them? m 1 + m 4 = 180 m 4 + m 2 = 180
36 Essential Question When two lines intersect, how do you know if two angles are congruent or supplementary and how do you use this information to find angle measures? 1.4 Perimeter and Area in the Coordinate Plane
37 Assignment
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