2.6 Prove Statements About Segments and Angles

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Presentation transcript:

2.6 Prove Statements About Segments and Angles

Objectives Write proofs involving segment and angle addition Write proofs involving segment and angle congruence

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, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof.

Two-Column Proof Statements Reasons Proof:

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

Your Turn: Prove the following. Prove: Given:

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

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.

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 .

Example 2: Prove the following. Prove: Given:

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

Your Turn: Prove the following. Prove: Given:

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

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