Question 1 Which congruence theorem proves that the following two triangles are congruent? Options: ASA Congruence AAS Congruence SSS Congruence HL Congruence Answer 1 The given triangles are congruent by the AAS (Angle-Angle-Side) Congruence theorem because: ∠ 𝑄 𝑅 𝑃 = ∠ 𝐵 𝐴 𝐶 ∠QRP=∠BAC ∠ 𝐴 𝐵 𝐶 = ∠ 𝑄 𝑃 𝑅 ∠ABC=∠QPR 𝐵 𝐶 = 𝑃 𝑄 BC=PQ Thus, △ 𝐴 𝐵 𝐶 ≅ △ 𝑄 𝑅 𝑃 △ABC≅△QRP by AAS Congruence. Answer: Option B (AAS Congruence). Question 2 How many sides does a hexagon have? Answer 1 The prefix "hex" means six in Greek. So, a hexagon has six sides. Question 3 Does congruence of triangles have the reflexive property, the symmetric property, and the transitive property? Answer 1 Step 1 Given: The congruence of triangles. Step 2 To identify whether the congruence of triangles has the reflexive property, symmetric property, and transitive property: Reflexive Property For any triangle △ 𝐷 𝐸 𝐹 △DEF: △ 𝐷 𝐸 𝐹 ≅ △ 𝐷 𝐸 𝐹 △DEF≅△DEF Hence, the congruence of triangles has the reflexive property. Symmetric Property If △ 𝐷 𝐸 𝐹 ≅ △ 𝑃 𝑄 𝑅 △DEF≅△PQR, then: △ 𝑃 𝑄 𝑅 ≅ △ 𝐷 𝐸 𝐹 △PQR≅△DEF Hence, the congruence of triangles has the symmetric property. Transitive Property If △ 𝐷 𝐸 𝐹 ≅ △ 𝑃 𝑄 𝑅 △DEF≅△PQR and △ 𝑃 𝑄 𝑅 ≅ △ 𝑋 𝑌 𝑍 △PQR≅△XYZ, then: △ 𝐷 𝐸 𝐹 ≅ △ 𝑋 𝑌 𝑍 △DEF≅△XYZ Hence, the congruence of triangles has the transitive property. Question 4 Select all the true statements based on the given diagram. Options: A. △ 𝐹 𝐺 𝐾 ≅ △ 𝐹 𝐽 𝐾 △FGK≅△FJK B. ∠ 𝐺 𝐾 𝐻 ≅ ∠ 𝐽 𝐾 𝐻 ∠GKH≅∠JKH C. 𝐹 𝐺 = 𝐾 𝐺 FG=KG D. ∠ 𝐺 𝐹 𝐻 ≅ ∠ 𝐽 𝐹 𝐻 ∠GFH≅∠JFH E. △ 𝐺 𝐾 𝐻 ≅ △ 𝐽 𝐾 𝐻 △GKH≅△JKH F. 𝐹 𝐺 = 𝐽 𝐾 FG=JK Answer 1 Step 1 In △ 𝐹 𝐺 𝐻 △FGH and △ 𝐹 𝐽 𝐻 △FJH: 𝐹 𝐺 = 𝐹 𝐽 FG=FJ (Given) 𝐺 𝐻 = 𝐽 𝐻 GH=JH (Given) 𝐹 𝐻 FH is a common side. Thus, by SSS Congruence: △ 𝐹 𝐺 𝐻 ≅ △ 𝐹 𝐽 𝐻 △FGH≅△FJH Step 2 Using the congruence from Step 1: A. True, by SAS Congruence. B. True, ∠ 𝐺 𝐾 𝐻 ≅ ∠ 𝐽 𝐾 𝐻 ∠GKH≅∠JKH by definition. C. False, 𝐹 𝐺 ≠ 𝐾 𝐺 FG  =KG. D. True, ∠ 𝐺 𝐹 𝐻 ≅ ∠ 𝐽 𝐹 𝐻 ∠GFH≅∠JFH. E. True, △ 𝐺 𝐾 𝐻 ≅ △ 𝐽 𝐾 𝐻 △GKH≅△JKH. F. False, 𝐹 𝐺 ≠ 𝐽 𝐾 FG  =JK. Conclusion: The true statements are A, B, D, and E. Question 5 Prove: △ 𝐴 𝐵 𝐶 ≅ △ 𝐸 𝐷 𝐶 △ABC≅△EDC. Answer 1 Consider the relations between the sides: 𝐶 𝐴 ‾ ≅ 𝐶 𝐵 ‾ CA ≅ CB 𝐶 𝐷 ‾ ≅ 𝐶 𝐸 ‾ CD ≅ CE Using the vertical angle congruence theorem, we know: ∠ 𝐴 𝐶 𝐵 ≅ ∠ 𝐸 𝐶 𝐷 ∠ACB≅∠ECD Therefore, by the congruence theorem, △ 𝐴 𝐵 𝐶 ≅ △ 𝐸 𝐷 𝐶 △ABC≅△EDC. Hence, proved.