Proving Statements about Segments and Angles in Geometry

1. 2.6 Prove Statements about Segments and Angles2.7 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.

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 1 Prove the following: • Segment congruence is reflexive • Angle congruence is symmetric To do these proofs, you have to turn “congruence” into “equality” and then turn “equality” back into “congruence.” In either case, just apply the Definition of Congruent Segments or Angles.

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

9. Example 1b Given: Prove: 1. m<A=m<B Given 2. m<B=m<A Reflexive property of = 3. <B = <A Definition of congruency ῀

10. Example 2 Prove the following: If M is the midpoint of AB, then AB is twice AM and AM is one half of AB. Prove: AB = 2AM and AM = (1/2)AB Given: M is the midpoint of • M is midpt. Given • AM = MB Def, of midpt. • AM = MB Def. of Cong. • AM + MB = AB Seg. Add. Post. • AM + AM = AB Sub. • 2AM = AB Addition • AM = ½ AB Division ῀

11. 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? They would both be 900 They would be congruent

12. 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?

13. Linear Pair Postulate If two angles form a linear pair, then they are supplementary.

14. Example 4 Given: Prove: TRY before you click! • m<1 = 680 Given • <1 & <2 are linear pair Def. of Linear Pair • <1 & <2 are supp. Linear Pair Post. • m<1 + m<2 = 180 Def. of supp. • 68 + m<2=180 Subst. • m<2 = 112 Subtraction

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? They must be congruent

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: Just TRY IT! In your notebook

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.

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: Come on – You can DO THIS!! (In your notebook)