1. Section 2.6

2. A proof is a logical argument that shows a statement is true. The first proof format we will use is called a two- column proof. A two-column proof has numbered statements and corresponding reasons that show an argument in logical order. In a two-column proof, each statement in the left- hand column is either given information or the result of applying a known property or fact to statements already made. Each reason in the right- hand column is the explanation for the corresponding statement.

3. Use the diagram to prove m  1 = m  4. Given: m  2 = m  3, m  AXD = m  AXC Prove: m  1 = m  4 Example 1

4. StatementsReasons 1.m  2 = m  3, m  AXD = m  AXC 1.Given 2.m  AXD = m  2 + m  1 m  AXC = m  3 + m  4 2.  Add. Post. 3.m  2 + m  1 = m  3 + m  4 3.Subst. Prop. 4.m  1 = m  4 4.Subt. Prop.

5. A theorem is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs. The reason used in a proof can include definitions, properties, postulates and theorems. Theorems

6. Segment congruence is reflexive, symmetric, and transitive. Reflexive Property: Symmetric Property: Transitive Property: Congruence of Segments Theorem

7. Angle congruence is reflexive, symmetric, and transitive. Reflexive Property: Symmetric Property: Transitive Property: Congruence of Angles Theorem

8. Name the property illustrated by the statement. a.If  5   3, then  3   5 Symmetric Property b.If  H   T and  T   B, then  H   B. Transitive Property Example 2

9. In Section 2.6, most of the proofs involve showing that congruence and equality or equivalent. You may find that what you are asked to prove seems to be obviously true. It is important to practice writing these proofs so that you will be prepared to write more complicated proofs in later chapters.

10. Given: Ray BD bisects  ABC Prove: m  ABC = 2 ∙ m  1 Example 3

11. StatementsReasons 1.Ray BD bisects  ABC 1.Given 2.m  1 = m  22.Def. of  Bis. 3.m  2 + m  1 = m  ABC3.  Add. Post. 4.m  ABC = 2 ∙ m  1 4.Subst. Prop.

12. 1.Draw or copy diagrams and label given information to help develop proofs. 2.Write or copy the Given and Prove statements. 3.The first statement and reason pair you write is given information 4.The statements are based on facts that you know or on conclusions from deductive reasoning. Writing a Two-Column Proof

13. 5.The reasons are definitions, postulates, properties, or proven theorems that allow you to state the corresponding statement. 6.Always remember to number the statements and reasons. 7.Remember to give a reason for the last statement which will always be the Prove Statement.

14. Write a two-column proof for the following statement. If you know that M is the midpoint of segment AB, N is the midpoint of segment CD, AB = CD, prove that AM = CN Example 4

15. Step 1: Draw a diagram from the previous statement.

16. Step 2: Write the Given and Prove statements from the diagram. Step 3: Write the two-column proof.

17. StatementsReasons 1.1.Given 2.AM = MB; CN = ND2.Def. of Midpt. 3.AM+ MB= AB CN + ND = CD 3.Seg. Add. Post. 4.AM + MB = CN + ND4.Subst. Prop. 5.AM = CN5.Subt. Prop.