Question 1 Conduct a formal proof to prove that the diagonals of an isosceles trapezoid are congruent. Given: Consider the isosceles trapezoid 𝐴 𝐵 𝐶 𝐷 ABCD where 𝐴 𝐵 = 𝐶 𝐷 AB=CD. Answer 1 To prove: 𝐴 𝐶 ≅ 𝐵 𝐷 AC≅BD Proof: Consider two triangles, △ 𝐴 𝐵 𝐶 △ABC and △ 𝐷 𝐶 𝐵 △DCB. Since 𝐴 𝐵 𝐶 𝐷 ABCD is an isosceles trapezoid, the base angles are congruent: ∠ 𝐴 𝐵 𝐶 ≅ ∠ 𝐷 𝐶 𝐵 ∠ABC≅∠DCB By the reflexive property, 𝐵 𝐶 = 𝐵 𝐶 BC=BC. Given that 𝐴 𝐵 = 𝐶 𝐷 AB=CD, we can use the SAS Postulate. Thus, △ 𝐴 𝐵 𝐶 ≅ △ 𝐷 𝐶 𝐵 △ABC≅△DCB by SAS. Therefore, 𝐴 𝐶 ≅ 𝐵 𝐷 AC≅BD, proving that the diagonals are congruent. Question 2 Choose the correct congruence theorem for triangles 𝐴 𝐵 𝐶 ABC and 𝐷 𝐶 𝐵 DCB. Options: a) Triangles 𝐴 𝐵 𝐶 ABC and 𝐷 𝐶 𝐵 DCB are congruent by the Side-Side-Side (SSS) Triangle Congruence Theorem. b) Triangles 𝐴 𝐵 𝐶 ABC and 𝐷 𝐶 𝐵 DCB are congruent by the Side-Angle-Side (SAS) Triangle Congruence Theorem. c) Triangles 𝐴 𝐵 𝐶 ABC and 𝐷 𝐶 𝐵 DCB are congruent by the Angle-Angle-Side (AAS) Triangle Congruence Theorem. d) Not enough information to determine if the triangles are congruent. Answer 1 The correct answer is: b) Triangles 𝐴 𝐵 𝐶 ABC and 𝐷 𝐶 𝐵 DCB are congruent by the Side-Angle-Side (SAS) Triangle Congruence Theorem, because two sides and the included angle are equal. Question 3 Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem. Answer 1 Proof of SAS: For the SAS theorem, two sides and the included angle should be equal in two triangles for the triangles to be congruent. In the given diagram: Two sides are provided, The included angle is shared (reflexive property), Therefore, there is enough information to conclude that the triangles are congruent by SAS. Question 4 Prove: The diagonals of a rectangle are equal in length. Given information: 𝐴 𝐵 𝐶 𝐷 ABCD is a rectangle. Answer 1 Concept used: Side-Angle-Side (SAS) congruence. Proof: Since 𝐴 𝐵 𝐶 𝐷 ABCD is a rectangle, it is also a parallelogram. Thus: 𝐴 𝐵 ≅ 𝐷 𝐶 AB≅DC (opposite sides of a parallelogram), 𝐵 𝐶 ≅ 𝐴 𝐷 BC≅AD (opposite sides of a parallelogram), ∠ 𝐴 𝐶 𝐵 = ∠ 𝐷 𝐶 𝐵 ∠ACB=∠DCB (all right angles in a rectangle are equal). By SAS Congruence: △ 𝐴 𝐶 𝐵 ≅ △ 𝐷 𝐶 𝐵 △ACB≅△DCB Therefore, 𝐴 𝐶 ≅ 𝐵 𝐷 AC≅BD. So, the diagonals 𝐴 𝐶 AC and 𝐵 𝐷 BD of a rectangle are equal in length. Question 5 Find the name of the corresponding parts in the congruent triangles and complete the congruence statement. Given: The congruence statement is △ 𝐹 𝐺 𝐻 ≅ △ 𝐿 𝑀 𝑁 △FGH≅△LMN. Answer 1 According to the question: 𝐹 𝐺 = 𝐿 𝑀 FG=LM, 𝐹 𝐻 = 𝐿 𝑁 FH=LN, 𝐺 𝐾 = 𝑀 𝑁 GK=MN. Thus, by SSS congruence, the complete statement is: △ 𝐹 𝐺 𝐻 ≅ △ 𝐿 𝑀 𝑁 △FGH≅△LMN