1. 2.8 Proving Angle Relationships

2. Objectives Write proofs involving supplementary and complementary angles Write proofs involving supplementary and complementary angles Write proofs involving congruent and right angles Write proofs involving congruent and right angles

3. Postulates Postulates 2.10 (Protractor Rule) All angles have measures between 0° and 180°. Postulate 2.11 (Angle Addition Postulate)  PQS iff m  PQR + m  RQS = m  PQS. R is in the interior of  PQS iff m  PQR + m  RQS = m  PQS. S P Q R

4. Postulates and Theorems Copy all of the Theorems from Section 2.8. Theorems 2.3 thru 2.13

5. If the second hand stops where the angle is bisected, then the angle between the minute and second hands is one-half the measure of the angle formed by the hour and minute hands, or. TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? Example 1:

6. By the Angle Addition Postulate, the sum of the two angles is 120, so the angle between the second and hour hands is also 60º. Answer: They are both 60º by the definition of angle bisector and the Angle Addition Postulate. Example 1:

7. Answer: 50 QUILTING The diagram below shows one square for a particular quilt pattern. If andis a right angle, find Your Turn:

8. Supplement Theorem Subtraction Property Answer: 14 form a linear pair andfindIfand Example 2:

9. Answer: 28 are complementary angles and. andIf find Your Turn:

10. Given: form a linear pair. Prove: In the figure, form a linear pair, and Prove that are congruent. and Example 3:

11. 1. Given 1. 3. Definition of supplementary angles 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Definition of congruent angles 6. Proof: Statements Reasons 2. Linear pairs are supplementary. 2. Example 3:  1 &  4 linear pair;

12. In the figure,  NYR and  RYA form a linear pair,  AXY and  AXZ form a linear pair, and  RYA and  AXZ are congruent. Prove that  RYN and  AXY are congruent. Your Turn:

13. Proof: Statements Reasons 1. Given 2. If two  s form a linear pair, then they are suppl.  s. 3. Given 4. 1. 2. 3. 4. linear pairs. Your Turn:

14. Substitution Add 2d to each side. Add 32 to each side. Divide each side by 3. If  1 and  2 are vertical angles and m  1 and m  2 find m  1 and m  2. Vertical Angles Theorem 11 22 Definition of congruent angles m1m1m2m2 Example 4:

15. Answer: m  1 = 37 and m  2 = 37 Example 4:

16. Answer: m  A = 52; m  Z = 52 find and If and are vertical angles andand Your Turn:

17. Assignment Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32 Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32 Pre-AP Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32, 34, 36, 38 Pre-AP Geometry: Pg. 111 – 113 #6, 16 – 24, 27 - 32, 34, 36, 38