Definitions  If . . . Then Statements If a ray bisects an angle, . . . then it divides the angle into two congruent angles. If a ray divides an angle into two congruent angles, . . . then the ray bisects the angle. If a line ( ray/segment/point) bisects a segment, . . . then it divides the segment into two congruent segments. If a line (ray/segment/point) divides a segment into two congruent segments, . . . then the line (ray/segment/point) bisects the segment. If a point divides a segment into two congruent segments, . . . then it is the midpoint of the segment. If a point is the midpoint of a segment, . . . then it divides the segment into two congruent segments.

Definitions  If . . . Then (cont.) Statements If two rays trisect an angle, . . . then they divides the angle into three congruent angles. If two rays divide an angle into three congruent angles, . . . then the rays trisect the angle.

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

2-5 Proving Angles Congruent Angle Pairs 60 Complementary Angles two angles whose measures add to 90. (Not necessarily adjacent.) Each is the complement of the other. 30 1 A B 2 Supplementary Angles two angles whose measures add to 180. (Not necessarily adjacent.) Each is the supplement of the other. D 120 60 5 6 C

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

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

2-5 Five Angle Theorems, cont. Congruent 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. C D 4 3 A 1 B A and 1 are complementary. B and 1 are complementary. Therefore, A  B. C and 3 are complementary. D and 4 are complementary. 3  4 Therefore, C  D.

2-5 Five Angle Theorems, cont. 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.

Proving the Five Theorems Vertical Angles Theorem 1 2 3

2 Congruent Supplements Theorem (Same Angle) 1 3

1 Congruent Supplements Theorem (Congruent Angles) 2 3 4

All Right Angles Congruent B All Right Angles Congruent A 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 2. mA = 90; mB = 90 3. mA = mB A z B Reasons 1. Given 2. Def. rt.  3. Substitution POE 4. Def. congruence

Angles Both Congruent and Supplementary are Right Angles X Given: X zY ; X supp.Y Prove: X and Y are right s Statements 1. X zY ; X supp.Y 2. mX= mY 3. mX+ mY=180 mX+ mX=180 2mX=180 mX=90 mY=90 X is a rt. ; Y is a rt.  Reasons Given Definition of congruence 3. Def. supp. s Substitution POE Combine like terms (simplify) Division POE Def. rt. 

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.

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

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

A 1 B X Y 3 2 4 Using the Theorems