1. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write proofs using geometric theorems

2. Premises in Geometric Arguments The following is a list of premises that can be used in geometric proofs: 1.Definitions and undefined terms 2.Properties of algebra, equality, and congruence 3.Postulates of geometry 4.Previously accepted or proven geometric conjectures (theorems)

3. Amazing hypothesis conclusion Usually we have to prove a conditional statement. Think of this proof as a maze, where the hypothesis is the starting point and the conclusion is the ending. p q

4. Amazing pq Your job in constructing the proof is to link p to q using definitions, properties, postulates, and previously proven theorems. p q

5. Example 1 Construct a two-column proof of: If m  1 = m  3, then m  DBC = m  EBA.

6. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons

7. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3

8. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given

9. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2

10. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property

11. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property 3. m  1 + m  2 = m  DBC

12. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property 3. m  1 + m  2 = m  DBC 3.Angle Addition Postulate

13. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property 3. m  1 + m  2 = m  DBC 3.Angle Addition Postulate 4. m  3 + m  2 = m  EBA

14. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property 3. m  1 + m  2 = m  DBC 3.Angle Addition Postulate 4. m  3 + m  2 = m  EBA 4.Angle Addition Postulate

15. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property 3. m  1 + m  2 = m  DBC 3.Angle Addition Postulate 4. m  3 + m  2 = m  EBA 4.Angle Addition Postulate 5. m  DBC = m  EBA

16. Example 1 Given: m  1 = m  3 Prove: m  DBC = m  EBA StatementsReasons 1. m  1 = m  3 1.Given 2. m  1 + m  2 = m  3 + m  2 2.Addition Property 3. m  1 + m  2 = m  DBC 3.Angle Addition Postulate 4. m  3 + m  2 = m  EBA 4.Angle Addition Postulate 5. m  DBC = m  EBA 5.Substitution Property

17. Two-Column Proof two-column given prove Notice in a two-column proof, you first list what you are given (hypothesis) and what you are to prove (conclusion). statements reasons The proof itself resembles a T-chart with numbered statements on the left and numbered reasons for those statements on the right. pq Before you begin your proof, it is wise to try to map out the maze from p to q.

18. Generic Two-Column Proof Given: ____________ Prove: ____________ StatementsReasons 1. 2. 3. Insert illustration here

19. 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.

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

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

22. Given: Prove: Given:M is the midpoint of AB Prove: AB is twice AM and AM is one half of AB. –M is the midpoint of AB –AM ≅ MB –AM=MB –AM+MB=AB –AM+AM=AB –2AM=AB –AM= AB/2 Given Definition of midpoint Def of congruence Segment Add Pos Substitution Simplify Division prop of equal

23. Assignment P. 116-119: 3,4, 10-13, 16, 21, 22 Finish for homework