Quadrilateral with vertices upper P, upper Q, upper R, upper S. Angles upper P and upper R are across from each other and marked congruent. Diagonal modifying above upper Q upper S with hash marks divides angle upper P upper Q upper R into two congruent angles. Sides modifying above upper P upper Q with hash marks and modifying above upper P upper R with hash marks are marked congruent.

Question

To prove quadrilateral PQRS is a kite, a student wants to show that PS¯¯¯¯¯¯¯≅RS¯¯¯¯¯¯¯ . The student will start by using the given information to prove triangles PQS and RQS are congruent. Which reason can the student use?

(1 point)
Responses

Side-Side-Side Triangle Congruence
Side-Side-Side Triangle Congruence

Angle-Angle-Angle Triangle Congruence
Angle-Angle-Angle Triangle Congruence

Angle-Side-Angle Triangle Congruence
Angle-Side-Angle Triangle Congruence

Angle-Side Triangle Congruence

Answer 1

To prove that quadrilateral \(PQRS\) is a kite by showing that triangles \(PQS\) and \(RQS\) are congruent, the student can use the following reasoning based on the provided information:

1. The angles \(P\) and \(R\) are congruent (given).
2. The diagonal \(QS\) is divided into two segments, with segments \(PQ\) and \(RS\) being marked congruent (given).
3. The angles \(PQS\) and \(RQS\) are congruent because \(QS\) is the same segment for both triangles.

Thus, the student has two angles and the included side congruent in triangles \(PQS\) and \(RQS\). Therefore, the student can use Angle-Side-Angle (ASA) Triangle Congruence to conclude that triangles \(PQS\) and \(RQS\) are congruent.

So, the most appropriate response would be:

Angle-Side-Angle Triangle Congruence.