Question

select the correct reasons to complete the proof. *hint draw the figure and mark the congruency. given: △pqr and △tsr are right - triangles. r is the mid - point of pt. pq = ts prove: △pqr≅△tsr statements reasons △pqr & △tsr are right △s given r midpoint pt given pq≅ts given pr≅rt x △pqr≅△tsr y

Explanation:

Step1: Definition of mid - point

A mid - point of a line segment divides the segment into two congruent segments. Since \(R\) is the mid - point of \(\overline{PT}\), by the definition of mid - point, \(\overline{PR}\cong\overline{RT}\).

Step2: Hypotenuse - Leg (HL) congruence theorem

We know that \(\triangle PQR\) and \(\triangle TSR\) are right triangles (\(\angle PQR=\angle TSR = 90^{\circ}\)), \(\overline{PQ}\cong\overline{TS}\) (given) and \(\overline{PR}\cong\overline{RT}\) (from the mid - point property). In right triangles, if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent by the Hypotenuse - Leg (HL) congruence theorem. So \(\triangle PQR\cong\triangle TSR\).

Answer:

• x: Definition of mid - point.
• y: Hypotenuse - Leg (HL) congruence theorem.