2. 6 Prove Statement about Segments and Angles 2 2.6 Prove Statement about Segments and Angles 2.7 Prove angle pair relationships

Theorem 2.1 Congruence of Segments Segment congruence is reflexive, symmetric and transitive. Reflexive – For any segment AB, AB AB. Symmetric – If AB CD, then CD AB. Transitive – If AB CD and CD EF, then AB EF.

Theorem 2.2 Congruence of Angles Angle congruence is reflexive, symmetric and transitive Reflexive – For any angle A, A A. Symmetric – If A B, then B A. Transitive – If A B and B C, then A C.

Example 1 Writing Two Column Proof: Given: Prove: Statements Reasons 1. 2. 3. 4. Given Definition of congruent angles Symmetric Property Definition of congruent angles

Example 2: Given: AC = AB + AB Prove: AB = BC Statements Reasons AB + BC = AC AB + AB = AB + BC AB = BC Given Segment Addition Postulate Transitive Property Subtraction Property

Example 3: Given: M is the midpoint of AB Prove: AB = 2AM Statements Reasons M is the midpoint of AB AM = MB AM + MB = AB AM + AM = AB 2AM = AB Given Definition of midpoint Definition of congruent segments Segment Addition Postulate Substitution Property Simplify or Collect Like Terms

Shopping Mall Example Walking down a hallway at the mall, you notice the music store is halfway between the food court and the shoe store. The shoe store is halfway between the music store and the bookstore. Prove that the distance between the entrances of the food court and music store is the same as the distance between the entrances of the shoe store and bookstore.

Shopping Mall Example STEP 1 Draw and label a diagram. STEP 2 Draw separate diagrams to show mathematical relationships. STEP 3 State what is given and what is to be proved for the situation. Then write a proof.

Shopping Mall Example Given Definition of midpoint B is the midpoint of AC . C is the midpoint of BD . PROVE: AB = CD STATEMENT REASONS 1. B is the midpoint of AC . C is the midpoint of BD . 1. Given 2. AB BC 2. Definition of midpoint 3. BC CD 3. Definition of midpoint 4. AB CD 4. Transitive Property of Congruence 5. AB = CD 5. Definition of congruent segments

Theorem 2.3 Right Angles Congruence Theorem – All right angles are congruent

Theorem 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. 1 2 3 If 1 and 2 are supplementary and 3 and 2 are supplementary, then 1 3

Theorem 2.5 Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. 4 5 6 If 4 and 5 are complementary and 6 and 5 are complementary, then 4 6

Extra Information Postulate 12 – Linear Pair Postulate If two angles form a linear pair, then they are supplementary. Theorem 2.6- Vertical Angles Congruence Theorem Vertical Angles are congruent.

Example 1 Identify the pairs of congruent angles. Explain how you know they are congruent. Answer:  

Example 2 Answers 15. m 3 = 38o 16. m DGE = 98o 17. m CGE = 136o 19. m AGC = 142o

Homework Assignment Textbook Page 108-109 # 3, 4, 16 Textbook Page 113 # 3, 5, 9, 11, 17, 19