Answer :

Final answer:

To prove triangle PQR is similar to triangle TSR, the AA similarity criterion is used, showing corresponding angles are equal and sides are proportional.

Explanation:

The question involves proving that two triangles are similar based on given information. To show that triangle PQR is similar to triangle TSR, we must prove that they have the same shape but not necessarily the same size. This can be accomplished by showing that all corresponding angles are equal and the sides are proportional using postulates and theorems such as AA (Angle-Angle) similarity, SSS (Side-Side-Side) similarity, and SAS (Side-Angle-Side) similarity criteria.

In this case, given that QR is parallel to PT and angle QPR is congruent to angle STR implies by the AA criterion that the triangles are similar. Therefore, using the properties of similar triangles, we can conclude triangle PQR is similar to triangle TSR.

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Rewritten by : Barada

Triangles PQR and TSR are similar by the AA similarity theorem, as it's given that they share one pair of congruent angles and both have right angles due to the lines being perpendicular.

To prove that triangles PQR and TSR are similar by the AA (Angle-Angle) similarity theorem, we first establish that ∠QPR≃∠STR as given.

Since the lines are perpendicular, this creation of two right angles further cements the AA condition for similarity.

Here's a step-by-step explanation:

QR⊥PT is given, which implies ∠QPT and ∠QRT are right angles.

Since all right angles are congruent, we have ∠QPT≃∠QRT.

∠QPR≃∠STR is given.

With two pairs of angles proved congruent (∠QPR≃∠STR and ∠QPT≃∠QRT), by the AA theorem, triangles PQR and TSR are similar triangles.