1. 2-5 Proving Angles Congruent Angle Pairs Vertical Angles two angles whose sides form two pairs of opposite rays. 1 2 3 4 Adjacent Angles two coplanar angles with a common side, a common vertex, and no common interior points. 5 6

2. 2-5 Proving Angles Congruent Angle Pairs Complementary Angles two angles whose measures add to 90 . (Not necessarily adjacent.) Each is the complement of the other. 1 2 Supplementary Angles two angles whose measures add to 180 . (Not necessarily adjacent.) Each is the supplement of the other. A B 60  30  56 C 60  D 120 

3. 2-5 Five Angle Theorems Vertical Angles Theorem Vertical angles are congruent. 1 2 3 4

4. 1 2

5. 1 2

6. 1 2

7. 30  AB E D C O

11. 2-5 Five Angle Theorems Congruent Supplements Theorem If two angles are supplements of the same angle, then the two angles are congruent.  A and  1 are supplementary.  B and  1 are supplementary. Therefore,  A   B. 3 C 4 D  C and  3 are supplementary.  D and  4 are supplementary.  3   4 Therefore,  C   D. A 1 B Congruent Supplements Theorem If two angles are supplements of congruent angles, then the two angles are congruent.

12. 2-5 Five Angle Theorems, cont.  A and  1 are complementary.  B and  1 are complementary. Therefore,  A   B. A 1 B  C and  3 are complementary.  D and  4 are complementary.  3   4 Therefore,  C   D. C D 4 3Congruent Complements Theorem If two angles are complements of the same angle, then the two angles are congruent. Congruent Complements Theorem If two angles are complements of congruent angles, then the two angles are congruent.

13. Right Angle Theorem All right angles are congruent. Congruent and Supplementary Theorem If two angles are congruent and supplementary, then each is a right angle. 2-5 Five Angle Theorems, cont.

14. Proving the Five Theorems Vertical Angles Theorem 1 2 3

15. 2 1 Congruent Supplements Theorem (Same Angle) 3

16. 2 1 Congruent Supplements Theorem (Congruent Angles) 3 4

17. Given:  A is a right angle.  B is a right angle. Prove:  A   B Statements 1.  A is a right angle;  B is a right angle Reasons 1.Given All Right Angles Congruent AB

18. Given:  X   Y ;  X supp.  Y Prove:  X and  Y are right  s Statements 1.  X   Y ;  X supp.  Y Reasons 1.Given Angles Both Congruent and Supplementary are Right Angles XY

19. Using the Theorems Now you can use these five theorems as part of other proofs. In the REASON column you can now write the short form abbreviations.

20. Complementary/Supplementary Proof A 1 B 2 3 4 C G E H F J D Statements 4.Substitution POC 4.  2 and  1 are supp. 1.Diagram;  3   1 2.  FJD is a straight angle 3.  2 and  3 are supp. Reasons 1.Given 2.Assumed from diagram 3.If two  s form a straight , then they are supp.

21. More Practice Statements X Y A O B Reasons 4.  AOX and  YOB are supp. 1.Diagram;  XOB   YOB 2.  AOB is a straight . 3.  AOX and  XOB are supp. 1.Given 4.Substitution POC 2.Assumed from diagram. 3.If two  s form a straight , then they are supp.

22. Using the Theorems A 1 B XY 3 2 4