Question

Given: ;$\mathbf{WP}\mathbf{=}\mathbf{ZP}$ .

Prove:$\angle \mathbf{WXY}\cong \angle \mathbf{ZYX}$ .

Short answer

It is given that$\mathbf{WP}\mathbf{=}\mathbf{ZP}$ and$\mathbf{PY}\mathbf{=}\mathbf{PX}$ .

In the triangles$\mathbf{WPX}$ and$\mathbf{ZPY}$ , it can be noticed that:

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

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

$\angle \mathbf{WPX}=\angle \mathbf{ZPY}\left(\mathbf{verticallyoppositeangles}\right)$

Therefore, in the triangles$\mathbf{WPX}$ and$\mathbf{ZPY}$ , it can be seen that $\mathbf{WP}\mathbf{=}\mathbf{ZP}$, $\angle \mathbf{WPX}\mathbf{=}\angle \mathbf{ZPY}$and$\mathbf{PY}\mathbf{=}\mathbf{PX}$ .

Therefore, the triangles$\mathbf{WPX}$ and $\mathbf{ZPY}$are the congruent triangles by using the SAS postulate.

Therefore, by using the corresponding parts of congruent triangles it can be obtained that $\angle \mathbf{WXP}\mathbf{=}\angle \mathbf{ZYP}$.

The angles opposite to the equal sides are also equal.

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

Therefore,$\angle \mathbf{PXY}\mathbf{=}\angle \mathbf{PYX}$ .

Therefore, it can be noticed that:

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

Therefore,$\angle \mathbf{WXY}\mathbf{=}\angle \mathbf{ZYX}$ .

Therefore,$\angle \mathbf{WXY}\cong \angle \mathbf{ZYX}$ .

Hence proved.

Step-by-step solution

- Observe the given diagram.

The given diagram is:

- Description of step.

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

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

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

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

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

Therefore, in the triangles$WPX$ and$ZPY$ , it can be seen that$WP=ZP$ , $\angle WPX=\angle ZPY$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 $\angle WXP=\angle ZYP$.

- Description of step.

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,$\angle PXY=\angle PYX$ .

Therefore, it can be noticed that:

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

Therefore,$\angle WXY=\angle ZYX$ .

Therefore,$\angle WXY\cong \angle ZYX$ .

Hence proved.