Question 1 What other information do you need in order to prove the triangles congruent using the SAS Congruence Postulate? Options: A) ∠ 𝐵 𝐴 𝐶 ≅ ∠ 𝐷 𝐴 𝐶 ∠BAC≅∠DAC B) 𝐴 𝐶 = 𝐵 𝐷 AC=BD C) ∠ 𝐵 𝐶 𝐴 = ∠ 𝐷 𝐶 𝐴 ∠BCA=∠DCA D) 𝐴 𝐶 = 𝐵 𝐷 AC=BD Answer 1 Step 1 Given: Two triangles △ 𝐴 𝐵 𝐶 △ABC and △ 𝐷 𝐴 𝐶 △DAC. Determine Needed Information To apply the SAS (Side-Angle-Side) congruence postulate, we need to prove two sides and the included angle of △ 𝐴 𝐵 𝐶 △ABC and △ 𝐷 𝐴 𝐶 △DAC are congruent. Step 2 The side 𝐵 𝐶 BC is congruent to 𝐶 𝐷 CD, and 𝐴 𝐶 AC is congruent to itself by the reflexive property. To apply SAS, we need the angle between these sides to be congruent. This means we need ∠ 𝐵 𝐴 𝐶 ≅ ∠ 𝐷 𝐴 𝐶 ∠BAC≅∠DAC. Answer: Therefore, to prove the triangles are congruent using SAS, we need information ∠ 𝐵 𝐴 𝐶 ≅ ∠ 𝐷 𝐴 𝐶 ∠BAC≅∠DAC (Option A). Question 2 Choose all correct statements about triangles. Statements: Triangle ABC and DCB are congruent by the Angle-Angle Triangle Congruence theorem. Triangle ABC and BCD are congruent by the Angle-Side-Angle Triangle Congruence theorem. Triangle ABC and BCD are congruent by the Side-Side-Side Triangle Congruence theorem. Triangle ABC and DCB are congruent by the Side-Angle-Side Triangle Congruence theorem. There is not enough information to determine if the triangles are congruent. Answer 1 Step 1 Given: From the figure, we can see: 𝐴 𝐵 = 𝐶 𝐷 AB=CD 𝐴 𝐶 = 𝐵 𝐷 AC=BD 𝐵 𝐶 = 𝐵 𝐶 BC=BC Apply SSS Property Since all corresponding sides are congruent, △ 𝐴 𝐵 𝐶 ≅ △ 𝐷 𝐶 𝐵 △ABC≅△DCB by the Side-Side-Side (SSS) property. Conclusion Answer: The correct statements are: Triangle ABC and DCB are congruent by the Side-Angle-Side Triangle Congruence theorem. Triangle ABC and DCB are congruent by the Side-Side-Side Triangle Congruence theorem. Question 3 Is there an analog to the SSS triangle congruence theorem for quadrilaterals? Answer 1 Step 1 To investigate whether an SSS congruence theorem exists for quadrilaterals, we apply the concept of side congruence. Explanation The SSS congruence theorem states that if all 3 sides of one triangle are congruent to all 3 sides of another triangle, then the triangles are congruent. In quadrilaterals, having four congruent sides does not guarantee that the quadrilaterals are congruent because the angles between sides can vary significantly. Conclusion Answer: There is no analog to the SSS triangle congruence theorem for quadrilaterals. Question 4 Determine whether the congruence is true or false. 88 ≡ 5 ( mod 9 ) 88≡5(mod 9) Answer 1 Calculation To determine if 88 ≡ 5 ( mod 9 ) 88≡5(mod 9) is true, we check if the remainder of 88 88 divided by 9 9 is 5 5. Solution Compute 88 − 5 = 83 88−5=83. Divide 83 83 by 9 9: 83 / 9 = 9.22 83/9=9.22, which is not an integer. Conclusion Answer: 88 ≡ 5 ( mod 9 ) 88≡5(mod 9) is false because 88 − 5 88−5 is not divisible by 9 9.