1. Proving Angles Congruent
Geometry Honors

2. Vocabulary
Vertical Angles – two angles whose sides form two pairs of opposite rays. 1 2

3. Vocabulary
Adjacent Angles – two coplanar angles with a common side, a common vertex, and no common interior points. 1 2 1 2

4. Vocabulary
Complementary Angles – two angles whose measures have a sum of 90. 2 2 1 1

5. Vocabulary
Supplementary Angles – two angles whose measures have a sum of 180. 2 1 1 2

6. A picture is worth a 1000 words….
Conclusions can be drawn from pictures if you see…
Adjacent Angles
Adjacent Supplementary angles
Vertical Angles

7. Conclusions that CANNOT be drawn from pictures
Congruent Angles
Congruent Segments
Right Angles
Parallel Lines
Perpendicular Lines
All of these must be marked to declare.

8. What conclusions can you draw from this picture?3 4 2 5 1

9. Vocabulary Postulate– an accepted statement of fact.
Theorem– a proven statement.
Proof– the steps taken to show that a conjecture is true.

10. Theorems 1 2 Vertical Angles Theorem - Vertical angles are congruent.
1  2

11. Theorems
Congruent Supplements Theorem - If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. 2 1 3
If 1 is supplementary to 2, and 3 is supplementary to 2, then 1  3.

12. Theorems
Congruent Complements Theorem - If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. 2 1 3
If 1 is complementary to 2, and 3 is complementary to 2, then 1  3.

13. Theorems
If two angles are congruent and supplementary, then each is a right angle. 2 3

14. Theorems
All right angles are congruent. 2 3
2  3

15. What theorem did you use?
Applying Theorems
Find x. 4x
Answer
x = 5
3x + 5
What theorem did you use?
Theorem
Vertical Angle Theorem

16. What theorem did you use?
Applying Theorems
Find x. 3x
Answer
80 - x
x = 20
What theorem did you use?
Theorem
Vertical Angle Theorem

17. What reasons did you use?
Applying Theorems
Find x and y.
3x + 8
5x - 20
5x + 4y
Answer
x = 14
y = 15
What reasons did you use?
Reasons
Vertical Angle Theorem & Definition of Supplementary

18. Applying Theorems Find x. x = 9 54 4x What reason did you use? Answer
Definition of Complementary

19. Applying Theorems Find x. x = 10 12x - 15 3x + 45
Answer
What reason did you use?
Reason
Definition of Supplementary

20. What reasons did you use?
Applying Theorems
Find x. 3x 2x
Answer
x = 18
What reasons did you use?
Reasons
Vertical Angles Theorem &
Definition of Complementary

21. Applying Theorems Find x. 4x + 5 x = 11 3x + 8
Answer
x = 11
What reason did you use?
Reason
Definition of Complementary

22. What reasons did you use?
Applying Theorems
x + y + 5 2x
Find x and y. y
x = 35
y = 70
Answer
What reasons did you use?
Reason
Vertical Angles Theorem, Substitution, & Def. of Supplementary

23. A and B are supplementary.
Applying Theorems
A and B are supplementary.
mA = 3x + 12 and mB = 2x – 22. What is the measure of each angle?
Answer
x = 38
mA =126
mB = 54

24. Applying Theorems
A is half as large as its complement, B . Find the measure of each angle?
mA = 30
mB = 60
Answer