1. 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.

2. 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.

3. 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

4. 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

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

6. 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.

7. 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.

8. 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.

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

10. 2
Congruent Supplements Theorem (Same Angle) 1 3

11. 1
Congruent Supplements Theorem (Congruent Angles) 2 3 4

12. All Right Angles CongruentB
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

13. Angles Both Congruent and Supplementary are Right AnglesX
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. 

14. 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.

15. Complementary/Supplementary ProofD 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

16. 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

17. A 1 B X Y 3 2 4
Using the Theorems