Question

T is a point on the line PS produced of a parallelogram PQRS and QT intersects RS at V. Prove that ΔPQT ~ ΔRVQ.

Answer 1

Given:

PQRS is a parallelogram.

T lies on the line PS produced (extended beyond S).

QT intersects RS at V.

To prove: ΔPQT ~ ΔRVQ

Proof:

Step 1: Understand the figure and notation

Since PQRS is a parallelogram, opposite sides are parallel and equal:

PQ || SR and PS || QR.

Point T lies on the line PS extended beyond S.

QT intersects RS at V.

Step 2: Identify angles to prove similarity

We want to prove similarity by Angle-Angle (AA) criterion, so we need to show two pairs of corresponding angles are equal.

Step 3: Show ∠PQT = ∠RVQ

Since PQ || SR (opposite sides of parallelogram) and QT is a transversal.

∠PQT and ∠RVQ are alternate interior angles.

Therefore, ∠PQT = ∠RVQ.

Step 4: Show ∠QPT = ∠QRV

Since PS || QR (opposite sides of parallelogram) and QT is a transversal.

∠QPT and ∠QRV are alternate interior angles.

Therefore, ∠QPT = ∠QRV.

Step 5: Conclude similarity

Two pairs of corresponding angles are equal:

∠PQT = ∠RVQ

∠QPT = ∠QRV

By AA criterion, ΔPQT ~ ΔRVQ.