91Ó°ÊÓ

The given diagram is:

It is given that$WP=ZP$ and $PY=PX$.

In the triangles $WPX$and $ZPY$, it can be noticed that:

$WP=ZP\left(\text{given}\right)$

$PY=PX\left(\text{given}\right)$

$∠WPX=∠ZPY\left(\text{verticallyoppositeangles}\right)$

Therefore, in the triangles$WPX$ and$ZPY$ , it can be seen that$WP=ZP$ , $∠WPX=∠ZPY$and$PY=PX$ .

Therefore, the triangles$WPX$ and$ZPY$ are the congruent triangles by using the SAS postulate.

Therefore, by using the corresponding parts of congruent triangles it can be obtained that $∠WXP=∠ZYP$.

The angles opposite to the equal sides are also equal.

As,$PY=PX$ , therefore the angles opposite to the sides$PY$ and$PX$ are also equal.

Therefore,$∠PXY=∠PYX$ .

Therefore, it can be noticed that:

$\begin{array}{c}∠WXY=∠WXP+∠PXY\\ =∠ZYP+∠PXY\\ =∠ZYP+∠PYX\\ =∠ZYX\end{array}$

Therefore,$∠WXY=∠ZYX$ .

Therefore,$∠WXYâ‰\dots ∠ZYX$ .

Hence proved.