Question

a. \\( \triangle s q p \cong \triangle r t q \\)
b. \\( \triangle s q p \cong \triangle t r q \\)
c. \\( \triangle p q s \cong \triangle t q r \\)
d. \\( \triangle p q s \cong \triangle t r q \\)

1. select the congruence theorem that proves the triangles are congruent:

a. sas
b. asa
c. aas
d. not enough information

Explanation:

Step1: Analyze the first problem (triangle congruence notation)

In triangle congruence, the order of the letters matters. We look at the corresponding parts. In the figure, we can see that \( \triangle PQS\) and \( \triangle TRQ\) have \(PQ = TR\) (marked with one tick), \(QS=RQ\) (marked with two ticks), and \( \angle PQS=\angle TRQ\) (vertical angles). So \( \triangle PQS\cong\triangle TRQ\) by SAS (Side - Angle - Side) congruence.

Step2: Analyze the second problem (congruence theorem)

We have two sides (marked with ticks) and the included angle (the angles between the two sides, which are equal as they are marked as congruent) in the two triangles. The SAS (Side - Angle - Side) congruence theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

For the first problem: D. \( \triangle PQS\cong\triangle TRQ\)
For the second problem: A. SAS