1. Warm Up • Why does this proof reach a false conclusion? a = b Given a² = ab Multi. Prop. a² + a² = a² + ab Add. Prop. 2a² = a² + ab Simplify 2a² – 2ab = a² + ab – 2ab Subt. Prop. 2a² – 2ab = a² – ab Simplify 2(a² – ab) = 1(a² – ab) Dist. Prop. 2 = 1 Div. Prop.

2. Geometry Segment and Angle Proofs

3. Learning Outcomes • I will be able to write a two-column proof for segment theorems. • I will be able to write a two-column proof for angle theorems.

4. Vocabulary • A theorem is a true statement that follows as a result of other true statements. • A two-column proof is a type of proof written as numbered statements and reasons that show the logical order of an argument. • A paragraph proof is a type of proof written in paragraph form. • A flow proof is a type of proof that uses arrows to show the flow of logical argument.

5. Paragraph Proof Example

6. Flow Proof Example

7. Two-column proof example

8. Steps of a proof • State the Given(s) • Translate The Given _ • Glean from picture _ • Combine _ • Check for Algebra • Translate back to prove statement • Given • Definition (usually congruence) • Properties and theorems • Substitution or transitive property • Algebraic properties • Definition (usually congruence)

9. Geometry Proofs • Brainstorm of ways to complete this proof with your partner.

10. 1st step: State the given • State the Given Given

11. 2nd step: Translate Given • Translate the Given: Given FR = AN definition of congruence

12. 3rd Step: Glean from Picture • Glean from picture Given FR = AN definition of congruence FR + RA = FA Segment Addition RA + AN = RN Postulate

13. 4th Step: Combine Combine using transitive property or substitution Given FR = AN definition of congruence FR + RA = FA Segment Addition RA + AN + RN Postulate FR + RA = FA Substitution RA + FR = RN FA = RN Transitive Property

14. 5th Step: Look for algebra Given FR = AN definition of congruence FR + RA = FA Segment Addition RA + AN + RN Postulate FR + RA = FA Substitution RA + FR = RN FA = RN Transitive Property

15. 6th step: Translate to prove statement Given FR = AN definition of congruence FR + RA = FA Segment Addition RA + AN = RN Postulate FR + RA = FA Substitution RA + FR = RN FA = RN Transitive Property Definition of Congruence

16. Common Segment Proofs

17. Common Segment Proofs

18. Linear pair postulate

19. Vertical Angle Theorem • Prove that angles 1 and 3 are congruent or that angles 2 and 4 are congruent.

20. Congruent supplements theorem • If two angles are supplementary to the same angle, then the two angles are congruent.

21. Individual practice