1. Mrs. McConaughyGeometry1 Introduction to Proof: During this lesson, we will:  Identify angles as adjacent or vertical  Identify supplementary and complementary angles and find their measures

2. Mrs. McConaughyGeometry2 Introduction to Proof: Part I Types of Angles

3. Mrs. McConaughyGeometry3 Review: Classifying Angles By Their Measures Recall, the degree measure, m, of an angle must be 0 > m ≥ 180. Angles can be classified into four categories by their measures: Acute Obtuse Right Straight

4. Mrs. McConaughyGeometry4 Classifying Angles By Their Position With Respect to Each Other Adjacent angles :_________________ ______________________________ ______________________________ two coplanar angles with a common side, a common vertex, and no common interior points 1 2 2 4 6 5 7 8 Which angles are adjacent to one another?

5. Mrs. McConaughyGeometry5 Classifying Angles By Their Position With Respect to Each Other Vertical angles: _________________ ______________________________ ______________________________ ______________________________ nonadjacent angles formed by intersecting lines. Vertical angles share a common vertex and have sides which are opposite rays. Which angles form vertical angle pairs? 1 4 2 3 5 3 7 9

6. Mrs. McConaughyGeometry6 Introduction to Proof: Part II Complementary & Supplementary Angles During this lesson, you will:  identify supplementary and complementary angles  determine the measures of supplementary and complementary angles

7. Mrs. McConaughyGeometry7 Definitions: Supplementary Angles and Linear Pairs Two angles are supplementary if the sum of their measures is 180 degrees. Each angle is called a supplement of the other. If the angles are adjacent and supplementary, they are called a linear pair.

8. Mrs. McConaughyGeometry8 Supplementary Angles and Linear Pairs m < 1 + m < 2 = 180 < 1 supplements < 2 < 1 is a supplement of < 2 m< PQS + m <SQR = 180 < PQS and < SQR are a linear pair Alert! Supplementary angles do not have to be adjacent. If they are adjacent, then the sides of the two angles which are not the common side form a straight angle. 1 2 m< GHJ + m <JHI = 180 < GHJ and < JHI are a linear pair

9. Mrs. McConaughyGeometry9 Example 1 Which are measures of supplementary angles? 30 ° and 160° 103° and 67° 86° and 94° 86° and 94° 180

10. Mrs. McConaughyGeometry10 Definition: Complementary Angles Complementary angles are related to right angles. Definition: Complementary Angles Two angles are complementary if the sum of their measures is 90 degrees. Each angle is called a complement of the other.

11. Mrs. McConaughyGeometry11 Complementary Angles Complementary angles do not have to be adjacent. If they are adjacent, then the sides of the two angles which are not the common side form a right angle.

12. Mrs. McConaughyGeometry12 Example 2 Find the measure of a complement of each angle, if possible. Find the measure of a supplement. Angle Measure ComplementSupplement 60° 95° m°90 - m180 - m 90180 ??

13. Mrs. McConaughyGeometry13 Example 3 Find the measure of an angle if its measure is 60° more than its supplement. m = 180 – m + 60 Alert! We will use the m, 90 - m, and 180 – m to solve problems about angles.

14. Mrs. McConaughyGeometry14 Example 4 Find the measure of an angle if its measure is twice that of its supplement.

15. Mrs. McConaughyGeometry15 Example 5 Find the measure of an angle if its measure is 40 less than four times the measure of its complement. measure is 40 less than four times the measure of its complement. m = 4 (90 – m) - 40

16. Mrs. McConaughyGeometry16 Final Checks for Understanding Which are measures of complementary angles?…supplementary angles?...neither? 1. 60° & 30° 2. 130° & 50°3. 114° & 66° 4. 92° & 2° 5. 53° & 47°6. 87° & 87°7. 45° & 45° 8. 26° & 154°

17. Mrs. McConaughyGeometry17 Final Checks for Understanding What is the measure of a complement of each angle whose measure is given? 1. 45°2. 20°3. 78°4. 46°5. 22.5 6. (m-5)°

18. Mrs. McConaughyGeometry18 Final Checks for Understanding Translate words mathematical symbols Complementary ___________________ Supplementary __________________ “a more than b” _______________________ “a less than b” _______________________

19. Mrs. McConaughyGeometry19 Homework Complementary & Supplementary Angles WS