1. Geometry Lesson 1.6 Angle Pair Relationships

2. Objectives Students will be able to: Define: vertical angles, linear pair, complementary angles, supplementary angles Identify vertical angles and linear pairs in a diagram Find angle measures directly or using algebraic expressions Identify complementary and supplementary angles in a diagram Find the measure of a complement or supplement

3. Warm-Up: Angle Types (review) Name the type of angle and state how many degrees it can have Right angle (exactly 90°) Acute angle < 90° Obtuse angle (> 90° and < 180°) Straight angle (exactly 180°)

4.  1 and  3 are vertical angles  2 and  4 are vertical angles 1. Vertical Angles Vertical Angles Vertical Angles: Two angles whose sides form two pairs of opposite rays not In this context, “vertical” means “shared vertex”, not “straight up” 1 3 42

5.  1 and  2 are a linear pair 2. Linear Pairs Linear Pair Linear Pair: Two adjacent angles whose non-common sides are opposite rays 1 2 Non-common sides are opposite rays Adjacent angles have a common side

6. Example 1: Identifying Angle Pairs Are  1 &  2 adjacent? Are  1 &  2 a linear pair? Are  3 &  4 a linear pair? Are  2 &  5 vertical angles? Are  1 &  4 vertical angles? Are  3 &  5 vertical angles? Yes: Common side Yes: Adjacent & opposite rays No: Adjacent, but not opposite rays Yes: Two pairs of opposite rays No: Sides are NOT opposite rays

7. Practice 1: Identify Angle Pairs Answer the questions for each figure (a)(b) No Yes No Yes No Yes No

8. 3. Properties of Angle Pairs The sum of the angle measures in a linear pair is always 180° Vertical Angles are always congruent (equal measures) 1 2 m  1 + m  2 = 180° 1 3 42 m  1 = m  3 m  2 = m  4

9. Simulations Vertical angles animation Linear pair animation

10. Example 2a: Finding Angle Measures 129° Linear pair: 51° + m  7 = 180° 103° Vertical angles are congruent 44° Linear pair: 136° + m  8 = 180° 53° Vertical angles are congruent

11. Example 2b: Finding Angle Measures This time, let’s use algebra… Find the value of x and use it to find the angle measures  FHI   GHJ (vertical  s) m  FHI  m  GHJ (7x – 25)° = (5x + 15)° 2x = 40  x = 20 Now, substitute x… m  FHI = 7(20) – 25 m  FHI = 115° m  GHJ = 5(20) + 15 m  GHJ = 115° 115°

12. Practice 2: Finding Angle Measures Find x or y and then evaluate the angle measures (a) (b)

13.  1 and  2 are complementary m  1 + m  2 = 90° 4. Complementary Angles Complementary Angles 90° Complementary Angles: Two angles whose measures total 90° 1 2 Adjacent 1 2 Non-adjacent or

14. 5. Supplementary Angles Supplementary Angles 180° Supplementary Angles: Two angles whose measures total 180° 5 6 AdjacentNon-adjacent 5 6 or  5 and  6 are supplementary m  1 + m  2 = 180°

15. Example 3a: Complements & Supplements  E is a complement of  F If m  E = 68°, find m  F m  E + m  F = 90° 68° + m  F = 90°  m  F = 22°  G is a supplement of  H If m  G = 152°, find m  H m  G + m  H = 180° 152° + m  H = 180°  m  H = 28° E F G H

16. Example 3b: Comp & Supp w/Algebra  A is supplementary to  B  A is complementary to  C m  A = x°; m  B = (x + 40)° m  C = (x – 50)° Find all angle measures A B A C m  A + m  B = 180° 2x + 40 = 180  x = 70 m  A = 70°m  B = (70 + 40)° = 110° m  C = (70 – 50)° = 20°

17. Practice 3: Comp & Supp  s (a)  A is a complement of  B and m  A = 81° Find m  B (b)  C is a supplement of  D and m  C = 27° Find m  D (c) Repeat example 3b with the following: m  A = x°; m  B = (2x)°; and m  C = (x – 30)°

18. Closure: Angle Pairs Angle measures in a linear pair add up to ________° Angle measures in vertical angles are _____________ Complementary angle measures add up to _______° Supplementary angle measures add up to _______° For some cool computer animations, go to http://www.mathopenref.com/tocs/anglestoc.html

19. Complementary Angles You’re sooo acute!!

20. Homework 1.6 w/s