Question

8. Given: PQR and TSR are right triangles,R is the midpoint of PT, PQ ≈ STProve: PQR ≈ TSR

Answer 1

Angle RPQ is equal to Angle RTS (given). R is the midpoint of QS; QR is equal to RS. Vertical opposite angles PRQ and SRT are also equal. Therefore, triangles PQR and TSR are congruent through the Side-Angle-Side (SAS) criterion.

Given that Angle RPQ equals Angle RTS, R is the midpoint of QS with QR equal to RS, and vertical opposite angles PRQ and SRT are also equal, the triangles PQR and TSR exhibit congruence by the Side-Angle-Side (SAS) criterion.

This congruence is established by the shared side QR equal to RS, the corresponding angles RPQ and RTS, and the fact that R is the midpoint of QS, forming the basis for a geometric proof.

Hence, the congruence of triangles PQR and TSR is logically concluded through the provided information and the SAS criterion in geometric reasoning.

Answer 2

Let's begin by identifying key information given to us:

Angle RPQ is equal to Angle RTS (given)

R is the midpoint of QS (given); QR is equal to RS

Based on the Side-Angle-Side (SAS) theorem, which states that if two sides and an included angle of a triangle is equal to that of another triangle, then, both are