Introduction to Proof:

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

2. Introduction to Proof: Part I Types of Angles Geometry

3. 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 Geometry

4. 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 Which angles are adjacent to one another? 1 2 2 4 6 5 7 8 Geometry

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

6. 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 Geometry

7. Definitions: Supplementary Angles and Linear Pairs Two angles are supplementaryif 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. Geometry

8. Supplementary Angles and Linear Pairs 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. m< PQS + m <SQR = 180 < PQS and < SQR are alinear pair 2 1 m < 1 + m < 2 = 180 < 1 supplements < 2 < 1 is a supplement of < 2 m< GHJ + m <JHI = 180 < GHJ and < JHI are alinear pair Geometry

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

10. Definition: Complementary Angles Complementary angles are related to right angles. Geometry

11. 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. Geometry

12. Example 2 ?? Find the measure of a complement of each angle, if possible. Find the measure of a supplement. 90 180 Geometry

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

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

15. 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 thanfour times the measure of its complement. m = 4 (90 – m)- 40 Geometry

16. Final Checks for Understanding Which are measures of complementary angles?…supplementary angles? ...neither? Geometry

17. Final Checks for Understanding What is the measure of a complement of each angle whose measure is given? Geometry

18. Final Checks for Understanding Translate words mathematical symbols Geometry

19. Homework Complementary & Supplementary Angles WS Geometry