Two Column Proofs

Honors Geometry Review - Two Column Proofs Name: KEY Date: Pd: 1. Given: 4 + 2(3x + 5) = 11 - x Prove: x = -3/7 Statement Justification 1. 4 + 2(3x + 5) = 11 - x 1. Given 2. 4 + 6x + 10 = 11 - x 2. Distributive Property 3. 6x + 14 = 11 - x 3. Simplify 4. 7x + 14 = 11 4. Addition POE 5. 7x = -3 5. Subtraction POE 6. x = -3/7 6. Division POE 2. Given: 3x^2 - 2x - 4 = 2x(x - 1) + 12 x < 0 Prove: x = -4 Statement Justification 1. 3x^2 - 2x - 4 = 2x(x - 1) + 12 1. Given 2. 3x^2 - 2x - 4 = 2x^2 - 2x + 12 2. Distributive Property 3. x^2 - 2x - 4 = -2x + 12 3. Subtraction POE 4. x^2 - 16 = 12 4. Addition POE 5. x^2 = 16 5. Addition POE 6. x = ±4 6. Def of Square Root 7. x = -4 7. Given x < 0 3. Given: ∠BAC is a right angle, ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are complementary. Statement Justification 1. ∠BAC is a right angle 1. Given 2. m∠BAC = 90° 2. Def of a right angle 3. m∠1 + m∠2 = m∠BAC 3. Angle Addition Postulate 4. m∠1 + m∠3 = 90° 4. Substitution POE 5. m∠2 ≅ m∠3 5. Given 6. m∠1 + m∠3 = 90° 6. Substitution POE 7. ∠1 and ∠3 are complementary 7. Def of complementary angles 4. Given: BX bisects < ABC, m< XBC = 45 Prove: < ABC is a right angle Statement Justification 1. BX bisects < ABC 1. Given 2. < XBA = < XBC 2. Def of a bisector 3. m< XBA = m< XBC 3. Def of congruent angles 4. m< XBC = 45 4. Given 5. m< XBA = 45 5. Substitution 6. m< XBA + m< XBC = m< ABC 6. Angle Addition Post. 7. 45 + 45 = m< ABC 7. Substitution 8. 90 = m< ABC 8. Simplify 9. < ABC is a right angle 9. Def of a right angle 5. Given: AD = BC, AD = 8, BC = 8, CD = 8 Prove: AD = CD Statement Justification 1. AD = BC = 8, CD = 8 1. Given 2. BC = CD 2. Def of congruency 3. AD = CD 3. Substitution POE 4. CD = 8 4. Substitution POE 5. AD = 8 5. Substitution POE 6. AD = CD 6. Def of congruency 6. Given: M is the midpoint of AB Prove: AM = 1/2 AB Statement Justification 1. M is the midpoint of AB 1. Given 2. AM = MB 2. Def of a midpoint 3. AM + MB = AB 3. Segment Addition Post. 4. AM + AM = AB 4. Substitution POE 5. 2AM = AB 5. Simplify 6. AM = 1/2 AB 6. Multiplication POE (or Division POE) 7. Given: 3x - 4 = 1/4(x + 12) Prove: x = 4 Statement Justification 1. 3x - 4 = 1/4(x + 12) 1. Given 2. 12x - 16 = x + 12 2. Distributive Property 3. 11x - 16 = 12 3. Subtraction POE 4. 11x = 28 4. Addition POE 5. x = 28/11 5. Multiplication POE (Division POE) 8. Given: Figure at the right... Prove: x = 9 Statement Justification 1. DF = EF, DE = 1/3x + 1, EF = 7 1. Given 2. DE + EF = DF 2. Segment Addition Postulate 3. 1/3x + 1 + 7 = 11 3. Substitution POE 4. 1/3x + 8 = 11 4. Simplify 5. 1/3x = 3 5. Subtraction POE 6. x = 9 6. Multiplication POE (Division POE) 9. Given: 2x^2 - 3x - 16 = -x(3x + x^2) Prove: x = 2 or x = -2 Statement Justification 1. 2x^2 - 3x - 16 = -x(3x + x^2) 1. Given 2. 2x^2 - 3x - 16 = -3x^2 - x^3 2. Distributive Property 3. 2x^3 - 3x^2 - 16 = -3x^2 - x^3 3. Simplify 4. 2x^3 - 16 = -x^3 4. Addition POE 5. 3x^3 - 16 = 0 5. Addition POE 6. 3x^3 = 16 6. Addition POE 7. x^3 = 16/3 7. Division POE 8. x = (16/3)^(1/3) 8. Def of a cube root 9. x = 2 or x = -2 9. Calculator