Question

given △abc≅△def, which of the following can be proven using cpctc? a. the areas of △abc and △def are identical. b. △abc and △def have equal perimeters. c. corresponding angles in △abc and △def are equal. d. the corresponding medians of △abc and △def are equal. which postulate can be used to prove that two triangles are congruent if two sides and the included angle are known? a. aas b. sss c. asa d. sas if triangles pqr and stu are congruent by the sas postulate, and pq = 9 cm, qr = 11 cm, and ∠pqr = 70 degrees, what is the length of st? a. 10 cm b. 12 cm c. 11 cm d. 9 cm

Explanation:

Step1: Recall CPCTC

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) states that if two triangles are congruent, their corresponding sides and corresponding angles are equal.

Step2: Analyze first - question options

Areas and perimeters are not directly proven by CPCTC. While corresponding medians are equal for congruent triangles, the most fundamental property proven by CPCTC is that corresponding angles (and sides) are equal. So for the first question, the answer is based on the definition of CPCTC for angles.

Step3: Recall triangle - congruence postulates

The AAS (Angle - Angle - Side) postulate requires two angles and a non - included side. SSS (Side - Side - Side) requires all three sides. ASA (Angle - Side - Angle) requires two angles and the included side. SAS (Side - Angle - Side) requires two sides and the included angle, which is what the second question asks for.

Step4: Use congruence property for third question

If \(\triangle PQR\cong\triangle STU\) by SAS and \(PQ = 9\) cm, then the corresponding side to \(PQ\) in \(\triangle STU\) is \(ST\). Since corresponding sides of congruent triangles are equal, \(ST=PQ = 9\) cm.

Answer:

1. c. Corresponding angles in \(\triangle ABC\) and \(\triangle DEF\) are equal.
2. d. SAS
3. d. 9 cm