Question

1. given △abc≅△def, which of the following can be proven using cpctc? a. △abc and △def have equal perimeters. b. the areas of △abc and △def are identical. c. corresponding angles in △abc and △def are equal. d. the corresponding medians of △abc and △def are equal. 5. which of the following statements is true if two triangles are congruent by asa? a. they have the same area but different perimeters. b. they have corresponding angles and included sides that are equal. c. they are similar, but not congruent. d. they have the same angles, but their side lengths differ. 6. what type of angle is critical in the sas rule? a. the angle opposite one side. b. the angle closest to the base. c. the largest angle in the triangle. d. the included angle between two sides.

Explanation:

Step1: Recall CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

CPCTC states that if two triangles are congruent, their corresponding sides and corresponding angles are congruent.

Step2: Analyze question 4

For $\triangle ABC\cong\triangle DEF$, by CPCTC, corresponding angles in $\triangle ABC$ and $\triangle DEF$ are equal. Perimeter and area equality are not direct results of CPCTC, and while corresponding medians are equal for congruent triangles, the most fundamental property proven by CPCTC is angle - side correspondence. So for question 4, the answer is c.

Step3: Recall ASA (Angle - Side - Angle) congruence criterion

The ASA congruence criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent. So for two triangles congruent by ASA, they have corresponding angles and included sides that are equal. This answers question 5, and the answer is b.

Step4: Recall SAS (Side - Angle - Side) congruence criterion

The SAS congruence criterion requires that two sides and the included angle between them of one triangle are equal to two sides and the included angle between them of another triangle. So the critical angle in the SAS rule is the included angle between two sides. This answers question 6, and the answer is d.

Answer:

1. c. Corresponding angles in $\triangle ABC$ and $\triangle DEF$ are equal.
2. b. They have corresponding angles and included sides that are equal.
3. d. The included angle between two sides.