91Ó°ÊÓ

The given diagram is:

The theorem $\mathbf{4}\mathbf{-}\mathbf{2}$states that if two angles of a triangle are congruent, then the sides opposite to those angles are congruent.

As, $∠3â‰\dots ∠4$,therefore it can be said that $∠PQXâ‰\dots ∠PTY$.

As, $∠5â‰\dots ∠6$, therefore by using the theorem $4-2$, it can be said that sides $PX$and $PY$are congruent sides.

Therefore, $PXâ‰\dots PY$.

From the given figure it can be noticed that the angles $∠PXQ$and $∠PXY$lies on the same straight line.

Therefore, the sum of the angles $∠PXQ$and $∠PXY$is role="math" localid="1648896543536" $180°$.

Therefore, $m∠PXQ+m∠PXY=180°$.

Now it can also be noticed that:

$\begin{array}{l}m∠PXQ+m∠PXY=180°\\ m∠PXQ+∠5=180°\\ m∠PXQ=180°−∠5\end{array}$

From the given figure it can be noticed that the angles $∠PYT$and $∠PYX$lies on the same straight line.

Therefore, the sum of the angles $∠PYT$and $∠PYX$is $180°$.

Therefore, $m∠PYT+m∠PYX=180°$.

Now it can also be noticed that:

$\begin{array}{l}m∠PYT+m∠PYX=180°\\ m∠PYT+∠6=180°\\ m∠PYT=180°−∠6\\ m∠PYT=180°−∠5(âˆ\mu ∠5=∠6)\end{array}$

Therefore, it can be noticed that $m∠PXQ=m∠PYT$.

That implies, $∠PXQâ‰\dots ∠PYT$

In the triangles $â–³PQX$and $â–³PTY$, it can be noticed that $∠PQXâ‰\dots ∠PTY,∠PXQâ‰\dots ∠PYT$and $PXâ‰\dots PY$.

Therefore, the triangles $â–³PQX$and $â–³PTY$are the congruent triangles by using the $AAS$postulate.

The postulate that proves the triangles$â–³PQX$ and$â–³PTY$ are congruent is $AAS$ postulate.