Geometry Proofs Key Concept: Properties of Equality & Distributive Property Let 𝑎, 𝑏, and 𝑐 be any real numbers. Addition Property If 𝑎 = 𝑏, then 𝑎 + 𝑐 = 𝑏 + 𝑐. Subtraction Property If 𝑎 = 𝑏, then 𝑎 − 𝑐 = 𝑏 − 𝑐. Multiplication Property If 𝑎 = 𝑏, then 𝑎 ∙ 𝑐 = 𝑏 ∙ 𝑐. !
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Division Property If 𝑎 = 𝑏 and 𝑐 ≠ 0, then = . !
Reflexive Property Symmetric Property Transitive Property Substitution Property Distributive Property* !
𝑎 = 𝑎. If 𝑎 = 𝑏, then 𝑏 = 𝑎. If 𝑎 = 𝑏 and 𝑏 = 𝑐, then 𝑎 = 𝑐. If 𝑎 = 𝑏, then 𝑏 can replace 𝑎 in any expression. Sum: 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 Difference: 𝑎 𝑏 − 𝑐 = 𝑎𝑏 − 𝑎𝑐 *Use the reason Distributive Property for combining like terms Key Concept: Properties of Congruence Reflexive Property 𝐴𝐵 ≅ 𝐴𝐵; ∠𝐴 ≅ ∠𝐴 Symmetric Property If 𝐴𝐵 ≅ 𝐶𝐷, then 𝐶𝐷 ≅ 𝐴𝐵. If ∠𝐴 ≅ ∠𝐵, then ∠𝐵 ≅ ∠𝐴. Transitive Property If 𝐴𝐵 ≅ 𝐶𝐷 and 𝐶𝐷 ≅ 𝐸𝐹, then 𝐴𝐵 ≅ 𝐸𝐹. If ∠𝐴 ≅ ∠𝐵 and ∠𝐵 ≅ ∠𝐶, then ∠𝐴 ≅ ∠𝐶. If ∠𝐵 ≅ ∠𝐴 and ∠𝐵 ≅ ∠𝐶, then ∠𝐴 ≅ ∠𝐶. Mini-­‐Proof 1: Justifying Steps When Solving an Equation What is the value of 𝑥 ? Justify each step. Statements ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary. 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = 180 2𝑥 + 30 + 𝑥 = 180 3𝑥 + 30 = 180 3𝑥 = 150 𝑥 = 50 Reasons Angles that form a linear pair are supplementary. Definition of supplementary angles OR Linear Pair Postulate Substitution Property Distributive Property Subtraction Property of Equality Division Property of Equality 1 Proof 1: Writing a Two-­‐Column Proof Write a two-­‐column proof. Given: 𝑚∠1 = 𝑚∠3 Prove: 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 Know: 𝑚∠1 = 𝑚∠3 Need: 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 Plan: Add 𝑚∠2 to both 𝑚∠1 and 𝑚∠3. The resulting angles will have equal measure. Statements Reasons 1. 𝑚∠1 = 𝑚∠3 1. Given 2. 𝑚∠2 = 𝑚∠2 2. Reflexive Property of Equality 3. 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2 3. Addition Property of Equality 4. 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐸𝐶 4. Angle Addition Postulate 𝑚∠3 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐵 5. 𝑚∠𝐴𝐸𝐶 = 𝑚∠𝐷𝐸𝐵 5. Substitution Property Proof 2: Writing a Two-­‐Column Proof Given: 𝐴𝐵 ≅ 𝐶𝐷 Prove: 𝐴𝐶 ≅ 𝐵𝐷 Statements 1.
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𝐴𝐵 ≅ 𝐶𝐷 𝐵𝐶 ≅ 𝐵𝐶 𝐴𝐵 = 𝐶𝐷 𝐴𝐵 + 𝐵𝐶 = 𝐶𝐷 + 𝐵𝐶 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 𝐵𝐶 + 𝐶𝐷 = 𝐵𝐷 6. 𝐴𝐶 = 𝐵𝐷 7. 𝐴𝐶 ≅ 𝐵𝐷 Reasons 1.
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Given Reflexive Property of Congruence Congruent segments have equal length Addition Property of Equality Segment Addition Postulate 6. Substitution Property 7. Segments with equal length are congruent 2 Proof 3: Vertical Angles Theorem Given: ∠1 and ∠3 are vertical angles Prove: ∠1 ≅ ∠3 Statements 1. ∠1 and ∠3 are vertical angles 2. ∠1 and ∠2 are supplementary ∠2 and ∠3 are supplementary 3. 𝑚∠1 + 𝑚∠2 = 180 𝑚∠2 + 𝑚∠3 = 180 4. 𝑚∠1 + 𝑚∠2 = 𝑚∠2 + 𝑚∠3 5. 𝑚∠1 = 𝑚∠3 6. ∠1 ≅ ∠3 Reasons 1. Given 2. Angles that form a linear pair are supplementary 3. The sum of the measures of supplementary angles is 180 4. Transitive Property of Equality 5. Subtraction Property of Equality 6. Angles with the same measure are congruent Proof 4: Proof using the Vertical Angles Theorem Given: ∠1 ≅ ∠4 Prove: ∠2 ≅ ∠3 Statements 1. ∠1 ≅ ∠4 2. ∠4 ≅ ∠2 3. ∠1 ≅ ∠2 4. ∠1 ≅ ∠3 5. ∠2 ≅ ∠3 Reasons 1. Given 2. Vertical angles are congruent (vertical angles theorem) 3. Transitive Property of Congruence 4. Vertical angles are congruent (vertical angles theorem) 5. Transitive Property of Congruence 3 Proof 5: Proof using the Vertical Angles Theorem Given: ∠1 ≅ ∠2 Prove: ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 Statements 1. ∠1 ≅ ∠2 2. ∠1 ≅ ∠3; ∠2 ≅ ∠4 3. ∠1 ≅ ∠4; ∠2 ≅ ∠3 4. ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4 Problem 3: Writing a Paragraph Proof Reasons 1. Given 2. Vertical angles are congruent (vertical angles theorem) 3. Transitive Property of Congruence 4. Transitive Property of Congruence Given: ∠1 and ∠3 are supplementary. ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠2 Proof: ∠1 and ∠3 are supplementary because it is given. So 𝑚∠1 + 𝑚∠3 = 180 by the definition of supplementary angles. ∠2 and ∠3 are supplementary because it is given, so 𝑚∠2 + 𝑚∠3 = 180 by the same definition. By the Transitive Property of Equality, 𝑚∠1 +
𝑚∠3 = 𝑚∠2 + 𝑚∠3. Subtract 𝑚∠3 from each side. By the Subtraction Property of Equality, 𝑚∠1 = 𝑚∠2. Angles with the same measure are congruent, so ∠1 ≅ ∠2. 4