1.4.4 Quiz - Week Four: Mathematical Proof

Identify the correct justification for each step, given AC intersects BE. m∠AFE= 40 and m∠CFD= 50

It is given that AC intersects BE. By the definition of vertical ∠s, ∠AFE and ∠BFC are vertical ∠s. By the Vertical ∠s Thm., ∠AFE≅∠BFC. By the definition of ≅∠s, m∠AFE=m∠BFC. It is given that m∠AFE=40°. So, m∠BFC=40° by the Subst. Property of =. By the ∠ Addition Post., m∠BFD=m∠BFC+m∠CFD. It is given that m∠CFD=50°. So, m∠BFD=40°+50° by the Subst. Property of =. Simplify to get m∠BFD=90°. Thus, ∠BFD is a right ∠ by the definition of right ∠s.

Identify the correct two-column proof for this plan. 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 = DE.

1. CD¯¯¯¯¯≅AC¯¯¯¯¯, BC¯¯¯¯¯≅CE¯¯¯¯¯: Given information 2. CD=AC, BC=CE: Definition of Congruent Segments 3. AC+CB=AB: Segment Addition Postulate 4. CD+CE=AB: Substitution Property of Equality 5. CD+CE=DE: Segment Addition Postulate 6. AB=DE: Transitive Property

Identify the correct justification for each step, given that ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary. 1. ∠1 and ∠2 are complementary ∠2 and ∠3 are complementary 2. m∠1 + m∠2 = 90° 3. m∠2 + m∠3 = 90° 4. m∠1 + m∠2 = m∠2 + m∠3 5. m∠1 = m∠3

1. Given 2. Def. of Comp. ∠s 3. Def. of Comp. ∠s 4. Trans. Prop. of = 5. Subt. Prop. of =

Identify the flowchart proof for the two-column proof. Given: m∠A = 18° and m∠B = 18° Prove: ∠A ≅ ∠B Two-Column Proof 1. m∠A = 18° (Given) 2. m∠B = 18° (Given) 3. m∠A = m∠B (Trans. Prop. of =) 4. ∠A ≅ ∠B (Def. of ≅ ∠s)

M∠A= 18° (Given) m∠B = 18° (Given) ------> M∠A = m∠B = 18° (Trans. Prop. of =) ∠A ≅ ∠B (Def. of ≅ ∠s)

Identify the property that justifies the statement. OP = RS, RS= OP

Sym. Prop. of ≅

Solve the equation. Write a justification for each step. t-13 ----- = -17 9

t-13 ----- = -17 (Given) 9 t-13= -63 t=-50

Identify the correct justification for each step, given that m∠1 ≅ m∠2, m∠3 = 110°, and ∠2 and ∠DBA are supplementary. 1. ∠2 and ∠DBA are supplementary 2. m∠2 + m∠DBA = 180° 3. m∠1 + m∠3 = m∠DBA 4. m∠2 + m∠1 + m∠3 = 180° 5. m∠1 ≅ m∠2 6. m∠2 + m∠2 + m∠3 = 180° 7. m∠3 = 110° 8. m∠2 + m∠2 + 110° = 180° 9. 2(m∠2) + 110° = 180° 10. 2(m∠2) = 70° 11. m∠2 = 35°

1. Given 2. Def. of Supp. ∠s 3. ∠ Add. Post. 4. Subst. Prop. of = 5. Given 6. Subst. Prop. of = 7. Given 8. Subst. Prop. of = 9. Simplify 10. Subt. Prop. of = 11. Div. Prop. of =

Solve the equation. Write a justification for each step. 2(x-4)=-4x +10

2(x-4)=-4x +10 2x-8= -4x+10 6x-8=10 6x=18 x=3

Identify the paragraph proof for the two-column proof. Given: m∠t = 3 ⋅ m∠s Prove: m∠s = 45° Two-Column Proof 1. m∠t = 3 ⋅ m∠s (Given) 2. ∠t and ∠s are supplementary (Lin. Pair Thm.) 3. m∠t + m∠s = 180° (Def. of Supp. ∠s) 4. 3 ⋅ m∠s + m∠s = 180° (Subst. Prop. of = ) 5. 4 ⋅ m∠s = 180° (Simplify.) 6. m∠s = 45° (Div. Prop. of = )

It is given that m∠t = 3 ⋅ m∠s. ∠t and ∠s are supplementary angles by the Linear Pair Theorem. By definition of supplementary angles, m∠t + m∠s = 180°. By Substitution Property of Equality, 3 ⋅ m ∠s + m∠s = 180°. By simplification, 4 ⋅ m ∠s = 180°. By Division Property of Equality, m∠s = 45°.

Write a paragraph proof to prove that ∠2 and ∠3 are complementary given that ∠1 and ∠2 are complementary, and m∠1 = m∠3.

It is given that ∠1 and ∠2 are complementary. By the definition of complementary angles, m∠1 + m∠2 = 90°. It is given that m∠1 = m∠3. So, by the Substitution Property of Equality, m∠3 + m∠2 = 90°. Thus, by the definition of complementary angles, ∠2 and ∠3 are complementary.