advertisement

2-6 Geometric Proof Homework 1. In a two column proof, you list the statements in the left column and the reasons in the right column. 2. A theorem is a statement you can prove. 3. Write a justification for each step, given that m∠A = 60° and m∠B = 2m∠A. 1. 2. 3. 4. 5. 6. Statements m∠A = 60°, m∠B = 2m∠A m∠B = 2(60°) m∠B = 120° m∠A + m∠B = 60° + 120° m∠A + m∠B = 180° ∠A and ∠B are supplementary Reasons 1. 2. 3. 4. 5. 6. Given Substitution Simplify Addition Property of Equality Simplify Definition of Supplementary Angles 4. Fill in the blanks to complete the two column proof. Given: ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are supplementary. 1. 2. 3. 4. 5. 6. Statements ∠2 ≅ ∠3 m∠2 = m∠3 b. ∠1 and ∠2 are supplementary angles m∠1 + m∠2 =180° m∠1 + m∠3 =180° d. ∠1 and ∠3 are supplementary 1. 2. 3. 4. 5. 6. Reasons Given a. Definition of Congruent Angles Linear Pair Theorem Definition of Supplementary Angles c. Substitution Definition of Supplementary Angles 5. Use the given plan to write a two-column proof. Given: X is the midpoint of ̅̅̅̅ , and Y is the midpoint of ̅̅̅̅ . Prove: ̅̅̅̅ ≅̅̅̅̅ ̅̅̅̅≅ ̅̅̅̅, and ̅̅̅̅≅ ̅̅̅̅ . Plan: By the definition of midpoint, ̅̅̅̅≅ ̅̅̅̅. Use the Transitive Property to conclude that Statements 1. X is the midpoint ̅̅̅̅ Y is the midpoint of ̅̅̅̅ ̅̅̅̅ 2. ̅̅̅̅ ≅ ̅̅̅̅ ̅̅̅̅ ≅ ̅̅̅̅ 3. ≅ ̅̅̅̅ Reasons 1. Given 2. Definition of midpoint 3. Transitive Property of Congruence 6. Write a justification for each step , given that ⃗⃗⃗⃗⃗ bisects ∠ABC and m∠XBC = 45°. 1. 2. 3. 4. 5. 6. 7. 8. 9. Statements BX bisects ∠ABC ∠ABX ≅ ∠XBC m∠ABX ≅ m∠XBC m∠XBC = 45° m∠ABX = 45° m∠ABX + m∠XBC = m∠ABC 45° + 45° = m∠ABC 90° = m∠ABC ∠ABC is a right angle Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9. Given Definition of Angle Bisector Definition of Congruent Angles Given Substitution Angle Addition Postulate Substitution Simplify Definition of Right Angles Fill in the blanks to complete each two column proof. 7. Given: ∠1 and ∠2 are supplementary, and ∠3 and ∠4 are supplementary. ∠2 ≅ ∠3 Prove: ∠1 ≅ ∠4 1. 2. 3. 4. 5. 6. 7. Statements ∠1 and ∠2 are supplementary ∠3 and ∠4 are supplementary a. m∠1 + m∠2 =180° m∠1 + m∠3 =180° m∠1 + m∠2 = m∠3 + m∠4 ∠2 ≅ ∠3 m∠2 = m∠3 c. m∠1 = m∠4 ∠1 ≅ ∠4 Reasons 1. Given 2. Definition of Supplementary Angles 3. 4. 5. 6. 7. Substitution Given Definition of Congruent Angles Subtraction Property of Equality Definition of Congruent Angles 8. Given: ∠BAC is a right angle. ∠2 ≅ ∠3 Prove: ∠1 and ∠3 are complementary. 1. 2. 3. 4. 5. 6. 7. 8. Statements ∠BAC is a right angle m∠BAC = 90° b. m∠1 + m∠2 = m∠BAC m∠1 + m∠2 =90° ∠2 ≅ ∠3 c. m∠2 ≅ m∠3 m∠1 + m∠3 =90° e. ∠1 and ∠3 are complementary Reasons 1. 2. 3. 4. 5. 6. 7. 8. Given a. Definition of right angles Angle Addition Postulate Substitution Given Definition of Congruent Angles d. Substitution Definition of Complementary Angles 9. Use the given plan to write a two-column proof. Given: ̅̅̅̅ ≅ ̅̅̅̅ , ̅̅̅̅ ≅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ≅ Prove: Plan: Use the definition of congruent segments to write the given information in terms of lengths. ̅̅̅̅ ≅ ̅̅̅̅. Then use the Segment Addition Postulate to show that AB = CD and thus Statements 1. BE ≅ CE DE ≅ AE 2. BE = CE DE = AE 3. AE + BE = AB CE + DE = CD 4. DE + CE = AB 5. AB = CD 6. AB ≅ CD Reasons 1. Given 2. Definition of congruent segments 3. Segment Addition Postulate 4. Substitution 5. Substitution 6. Definition of congruent segments 10. Given: ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. ∠3 ≅ ∠4 Prove: ∠1 ≅ ∠2 Plan: Since ∠1 and ∠3 are complementary and ∠2 and ∠4 are complementary, both pairs of angle measures add to 90°. Use substitution to show that the sums of both pairs are equal. Since ∠3 ≅ ∠ 4, their measures are equal. Use the Subtraction Property of Equality and the definition of congruent angles to conclude that ∠1 ≅ ∠2. 1. 2. 3. 4. 5. 6. Statements ∠1 and ∠3 are complementary ∠2 and ∠4 are complementary ∠3 ≅ ∠4 m∠1 + m∠3 =90° m∠2 + m∠4 =90° m∠1 + m∠3 = m∠2 + m∠4 m∠3 = m∠4 m∠1 = m∠2 ∠1 ≅ ∠2 11. m∠1 = 180° - 48° = 132° 12. m∠2 = 90° - 63° = 27° 13. m∠3 = 90° - 31° = 59° Reasons 1. Given 2. Definition of Complementary Angles 3. 4. 5. 6. Substitution Definition of Congruent Angles Subtraction Property of Equality Definition of Congruent Angles 16. 17. 18. 19. 20. Sometimes Sometimes Sometimes Never 4 + 5 + 8 − 5 = 180° 12 = 180° = 15° 21. 9 − 6 = 8.5 + 2 . 5 − 6 = 2 . 5 = 8 = 16 22. 4 + 3 + 6 = 90° 7 + 6 = 90° 7 = 84° = 12°