1. To write proofs using geometric theorems
2.6 Prove Statements about Segments and Angles 2.7 Prove Angle Pair Relationships
Objectives:
To write proofs using geometric theorems
To use and prove properties of special pairs of angles to find angle measurements

2. Thanks a lot, Euclid!
Recall that it was the development of civilization in general and specifically a series of clever ancient Greeks who are to be thanked (or blamed) for the insistence on reason and proof in mathematics.

3. Premises in Geometric Arguments
The following is a list of premises that can be used in geometric proofs:
Definitions and undefined terms
Properties of algebra, equality, and congruence
Postulates of geometry
Previously accepted or proven geometric conjectures (theorems)

4. Properties of Equality
Maybe you remember these from Algebra.
Reflexive Property of Equality
For any real number a, a = a.
Symmetric Property of Equality
For any real numbers a and b, if a = b, then b = a.
Transitive Property of Equality
For any real numbers a, b, and c, if a = b and b = c, then a = c.

5. Theorems of Congruence
Congruence of Segments
Segment congruence is reflexive, symmetric, and transitive.

6. Theorems of Congruence
Congruence of Angles
Angle congruence is reflexive, symmetric, and transitive.

7. Example 1a Given: Prove: 2. has length AB Statements Reasons 1.
2.Ruler Postulate
3. AB = AB
3.Reflexive Prop. of = 4.
4.Definition of Congruent Segments

8. Example 1b
Given:
Prove:

9. Example 2 Prove the following:
If M is the midpoint of AB, then AB is twice AM and AM is one half of AB.
Given: M is the midpoint of
Prove: AB = 2AM and AM = (1/2)AB

10. Example 3a
If there was a right angle in Denton, TX, and other right angle in that place in Greece with all the ruins (Athens), what would be true about their measures?

11. Right Angle Congruence Theorem
All right angles are congruent. Yes, it seems obvious, but can you prove it? What would be your Given information? What would you have to prove?

12. Example 3b
Given: < A and < B are right angles
Prove:

13. Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
Do we have to prove this?

14. Example 4
Given: Prove:

15. Congruent Supplements
Suppose your angles were numbered as shown. Notice angles 1 and 2 are supplementary. Notice also that 2 and 3 are supplementary. What must be true about angles 1 and 3?

16. Congruent Supplement Theorem
If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

17. Example 5 Prove the Congruent Supplement Theorem.
Given: < 1 and < 2 are supplementary
< 2 and < 3 are supplementary
Prove:

18. What to Prove
Notice that you can essentially have two kinds of proofs:
Proof of the Theorem
Someone has already proven this. You are just showing your peerless deductive skills to prove it, too.
YOU CANNOT USE THE THEOREM TO PROVE THE THEOREM!
Proof Using the Theorem (or Postulate)

19. Congruent Complement Theorem
If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
You’ll have to prove this in your homework.

20. Vertical Angle Congruence Theorem
Vertical angles are congruent.

21. Example 6 Prove the Vertical Angles Congruence Theorem.
Given: < 1 and < 3 are vertical angles
Prove:

22. Example 7
Given: Prove:

23. Example 8
Given: Prove: < 3 and < 4 are supplements

24. Example 9