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Proving Equivalence and Congruence in Geometry: A Two-Column Proofs Guide

This document presents a series of geometric proofs using the two-column proof format. It covers multiple scenarios including manipulating algebraic equations to demonstrate solutions, segment addition, and proving angle congruence. Each proof is structured with statements and corresponding reasons, highlighting properties such as the Distributive Property, Substitution, and the Segment Addition Postulate. The proofs aim to establish congruences and equality between geometric figures and angles, clarifying fundamental concepts in geometry.

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Proving Equivalence and Congruence in Geometry: A Two-Column Proofs Guide

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  1. Two-Column Proofs Given: 2x - 3 = 23 Prove: x = 116 Statements 1. 2x - 3 = 23 2. 3(2x - 3) = 2 3. 6x - 9 = 2 4. 6x = 11 5. x = 116 Reasons 1. Given 2. Multiplication POE 3. Distributive property 4. Addition POE 5. Division Property

  2. C B A X m Two-Column Proofs Given: A, B, C, X on line m as shownAC = BX Prove: AB = CX Statements 1. A, B, C, X on line m as shown 2. AC = AB + BC 3. BX = BC + CX 4. AC = BX 5. AB + BC = BC + CX 6. AB = CX Reasons 1. Given 2. Segment Addition Postulate 3. Segment Addition Postulate 4. Given 5. Substitution (steps 2, 3, 4) 6. Subtraction POE

  3. C X Y A B Two-Column Proofs Given: AX  BY XC  YC Prove: AC  BC • Reasons • 1. Given • 2. Definition of Congruence • 3. Segment Addition Postulate • 4. Substitution, steps 3 and 4 • 5. Substitution, steps 4 and 5. • 6. Definition of Congruence Statements 1. AX  BY; XC  YC 2. AX = BY; XC = YC 3. AX + XC = AC; BY + YC = BC 4. BY + YC = AC 5. AC = BC 6. AC  BC

  4. Two-Column Proofs M Given: mMBA = 84 mABO = 42 Prove: MBO  ABO O B • Statements • mMBA = 84; mABO = 42 • mMBA = mMBO + mABO • mMBA – mABO = mABO • 84 – 42 = mABO • 42= mABO • mABO = mMBO • ABO  MB0 • Reasons • 1. Given • 2. Angle Addition Postulate • Subtraction POE • Substitution POE • Combine like terms (simplify) • If two s have the same measure, then they are equal. • Definition of congruence A

  5. Two-Column Proofs M A B N C D

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