In the given diagram, we are told that PQ is perpendicular to QT, that ST is perpendicular to TQ, and that PQST= QRTV. We want to prove that △ VKR is an isosceles triangle. Let's highlight triangles △ PQR and △ STV on the diagram.
As a first step, we will start by proving that △ PQR and △ STV are similar triangles. Let's investigate the given information.
Since PQ ⊥ QT and ST ⊥ TV, we know that ∠ PQR and ∠ STV are right angles. Because all right angles are congruent, we can say that ∠ PQR ≅ ∠ STV. Moreover PQ and ST are the heights of the triangles, and QR and TV are their bases.
It is given that the ratios PQ:ST and QR:TV are equal. Therefore, the heights and the bases of triangles △ PQR and △ STV are proportional.
We know that △ PQR and △ STV have two proportional sides and a congruent included angle. By the Side-Angle-Side Similarity Theorem, triangles △ PQR and △ STV are similar. Since corresponding angles in similar triangles are congruent, we know that ∠ PRQ≅ ∠ SVT.
