Congruence #3 - Q&A | Answer Key

University:
University of Michigan
Course:
MATH 425 | Introduction to Probability
Academic year:

2024
Views:

409

Pages:

4
Author:

Shyla Terry

Question 1 To Find: The congruent corresponding parts and then another congruence statement for the polygons. Given: △ 𝐴 𝐵 𝐶 ≅ △ 𝐷 𝐸 𝐹 △ABC≅△DEF Answer 1 Since △ 𝐴 𝐵 𝐶 ≅ △ 𝐷 𝐸 𝐹 △ABC≅△DEF: From the figure: 𝐴 𝐵 ≅ 𝐷 𝐸 AB≅DE 𝐵 𝐶 ≅ 𝐸 𝐹 BC≅EF 𝐶 𝐴 ≅ 𝐷 𝐹 CA≅DF ∠ 𝐴 𝐵 𝐶 ≅ ∠ 𝐷 𝐸 𝐹 ∠ABC≅∠DEF ∠ 𝐵 𝐶 𝐴 ≅ ∠ 𝐸 𝐹 𝐷 ∠BCA≅∠EFD ∠ 𝐶 𝐴 𝐵 ≅ ∠ 𝐹 𝐷 𝐸 ∠CAB≅∠FDE Another congruence statement: △ 𝐵 𝐶 𝐴 ≅ △ 𝐸 𝐹 𝐷 △BCA≅△EFD Question 2 To show that two sides are congruent. Given information: ∠ 𝐻 𝐺 𝐽 ≅ ∠ 𝐾 𝐽 𝐺 ∠HGJ≅∠KJG ∠ 𝐾 𝐺 𝐽 ≅ ∠ 𝐿 𝐻 𝐺 ∠KGJ≅∠LHG Answer 1 Formula used: The property used here is: Two triangles are congruent by the ASA (Angle-Side-Angle) congruence rule. Proof: It is given that: ∠ 𝐻 𝐺 𝐽 ≅ ∠ 𝐾 𝐽 𝐺 ∠HGJ≅∠KJG ∠ 𝐾 𝐺 𝐽 ≅ ∠ 𝐿 𝐻 𝐺 ∠KGJ≅∠LHG By the reflexive property, 𝐺 𝐽 ≅ 𝐺 𝐽 GJ≅GJ. By the ASA Congruence rule, we get: △ 𝐺 𝐻 𝐽 ≅ △ 𝐿 𝐾 𝐺 △GHJ≅△LKG Therefore, as corresponding parts of congruent triangles are congruent, 𝐻 𝐺 ≅ 𝐾 𝐽 HG≅KJ. Question 3 To find: The congruent criteria used to prove that the triangles are congruent. Given information: 𝑊 𝑈 ≅ 𝑍 𝑉 WU≅ZV and 𝑊 𝑋 ≅ 𝑌 𝑍 WX≅YZ ∠ 𝑈 ∠U and ∠ 𝑉 ∠V are right angles. Answer 1 Consider △ 𝑊 𝑈 𝑌 △WUY and △ 𝑍 𝑉 𝑋 △ZVX. Given: 𝑊 𝑈 ≅ 𝑍 𝑉 WU≅ZV (Given) ∠ 𝑈 = ∠ 𝑉 = 9 0 ∘ ∠U=∠V=90 ∘ (right angles) 𝑊 𝑋 ≅ 𝑌 𝑍 WX≅YZ (Given) Adding 𝑋 𝑌 XY to both sides: 𝑊 𝑋 + 𝑋 𝑌 = 𝑌 𝑍 + 𝑋 𝑌 WX+XY=YZ+XY So, 𝑊 𝑌 ≅ 𝑋 𝑍 WY≅XZ Therefore: △ 𝑊 𝑈 𝑌 ≅ △ 𝑍 𝑉 𝑋 ( RHS congruence criteria ) △WUY≅△ZVX(RHS congruence criteria) Thus, △ 𝑊 𝑈 𝑌 △WUY and △ 𝑍 𝑉 𝑋 △ZVX are congruent triangles. Question 4 Prove: In the following figure, triangles 𝐴 𝐷 𝐶 ADC and 𝐵 𝐷 𝐶 BDC are congruent. Given: 𝐴 𝐶 = 𝐵 𝐶 AC=BC and 𝐷 D is the midpoint of 𝐴 𝐵 AB. Answer 1 (SSS) - If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent. Statement and Reason: 𝐴 𝐶 = 𝐵 𝐶 AC=BC (Provided) 𝐴 𝐷 = 𝐷 𝐵 AD=DB (since 𝐷 D is the midpoint of 𝐴 𝐵 AB) 𝐶 𝐷 = 𝐶 𝐷 CD=CD (Reflexive property) Thus, △ 𝐴 𝐷 𝐶 ≅ △ 𝐵 𝐷 𝐶 △ADC≅△BDC by the SSS Congruence property. Question 5 Evaluate whether the provided data is sufficient to demonstrate △ 𝑊 𝑋 𝑍 ≅ △ 𝑌 𝑍 𝑋 △WXZ≅△YZX using the SAS Congruency Theorem. If so, provide evidence; if not, explain why. Answer 1 Calculation: Using the HL Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, the two triangles are congruent. From the given figure, the triangles △ 𝑊 𝑋 𝑍 △WXZ and △ 𝑌 𝑍 𝑋 △YZX are right triangles with: 𝑊 𝑋 ≅ 𝑌 𝑍 WX≅YZ 𝑋 𝑍 ≅ 𝑍 𝑋 XZ≅ZX (by Reflexive Property) By the HL Congruence Theorem, we conclude that: △ 𝑊 𝑋 𝑍 ≅ △ 𝑌 𝑍 𝑋 △WXZ≅△YZX