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2.6 Prove Statements About Segments and Angles

This guide provides comprehensive instructions for writing geometric proofs involving segment and angle addition, as well as congruence. Specifically, it introduces the two-column proof format, where statements and corresponding reasons are aligned for clarity. Essential properties, including reflexive, symmetric, and transitive properties of congruence for segments and angles, are discussed in detail. Examples and practice problems are included to reinforce understanding and application of these concepts in geometry.

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2.6 Prove Statements About Segments and Angles

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  1. 2.6 Prove Statements About Segments and Angles

  2. Objectives • Write proofs involving segment and angle addition • Write proofs involving segment and angle congruence

  3. Two-Column Proof Recall what a two-column proof was… • Two-Column Proof – A proof format used in geometry in which an argument is presented with two columns, statementsand reasons, to prove conjectures and theorems are true. Also referred to as a formal proof.

  4. Proof: Statements Reasons Two-Column Proof

  5. Prove the following. Given: PR = QS Prove: PQ = RS Proof: Statements Reasons 1. 1. Given PR = QS 2. 2. Subtraction Property PR – QR = QS – QR 3. 3. Segment Addition Postulate PR – QR = PQ; QS – QR = RS 4. 4. Substitution PQ = RS Example 1:

  6. Prove the following. Given: Prove: Your Turn:

  7. Proof: Statements Reasons 1. 1. Given AC = AB, AB = BX 2. 2. Transitive Property AC = BX CY = XD 3. 3. Given 4. AC + CY = BX + XD 4. Addition Property 5. 5. Segment Addition Property AC + CY = AY; BX + XD = BD 6. 6. Substitution AY = BD Your Turn:

  8. Congruence of Segments Theorem 2.1 (Congruence of Segments) Congruence of segments is reflexive, symmetric, and transitive. Reflexive Property: AB AB Symmetric Property: If AB  CD, then CD  AB. Transitive Property: If AB  CD and CD  EF, then AB  EF.

  9. Congruence of Angles Theorem 2.2 (Congruence of Angles) Congruence of angles is reflexive, symmetric, and transitive. Reflexive Property:A  A Symmetric Property: If A  B , then B  A Transitive Property: If A  B and B  C , then A  C .

  10. Prove the following. Given: Prove: Example 2:

  11. Proof: Statements Reasons 1. Given 1. 2. Definition of congruent segments 2. 3. 3. Given 4. Transitive Property 4. 5. Transitive Property 5. Example 2:

  12. Prove the following. Given: Prove: Your Turn:

  13. Statements Reasons 1. 1. Given 2. 2. Transitive Property 3. 3. Given 4. 4. Transitive Property 5. 5. Symmetric Property Your Turn: Proof:

  14. Assignment • Geometry: Pg. 116 – 119 #3 – 13, 16, 21, 22

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