EXERCISE 14.2

In the given figure, $\mathrm{ABCD}$ is a rectangle and $\mathrm{AP}=\mathrm{DQ}$ and $\mathrm{AR}=\mathrm{PD}$. Prove that $\mathrm{PQ}=\mathrm{PR}$.

In the given figure, $\mathrm{QR}=\mathrm{RS}, \mathrm{PQ}=\mathrm{TS} \text{and} \angle \mathrm{Q}=\angle \mathrm{S}$. Prove that $\mathrm{PR}=\mathrm{TR}$.

In the given figure, $\mathrm{O}$ is the midpoint of $\mathrm{BD}$. Find how many pairs of congruent triangles are there. Also state the criterion of congruency for each pair.

In the figure, $\mathrm{ABCD}$ is a rectangle and $\mathrm{AC}=\mathrm{EC}$ and $\mathrm{D}$ is the mid point of $\mathrm{AE}$. Find how many pairs of congruent triangles are there, Write the congruence of each pair in a symbolic form.

In $△\mathrm{PQR} \text{and} △\mathrm{XYZ}$, $\angle \mathrm{P}=\angle \mathrm{X}$ and $\angle \mathrm{R}=\angle \mathrm{Z}$ and $\mathrm{PQ}=\mathrm{XY}$. Are the triangles congruent? Give reasons.

Using the suitable congruence criterion for triangles, show that the opposite angles of a rhombus are equal.

In $△\mathrm{ABC}, \mathrm{P}$ is the mid point of $\mathrm{BC}\hspace{0.17em}\text{and} \mathrm{PF}\perp \mathrm{AB} \text{and} \mathrm{PE}\perp \mathrm{AC}$ such that $\mathrm{PE}=\mathrm{PF}$. Prove that $△\mathrm{ABC}$ is an isosceles triangle.

In the figure given below, $\mathrm{AP} \text{and} \mathrm{DQ}$ are the angle bisector of $\angle \mathrm{A} \text{and} \angle \mathrm{D}$ respectively, and $△\mathrm{ABP}\cong \angle \mathrm{DEQ}$. If $\mathrm{AC}=\mathrm{DF}$, then is it true that $△\mathrm{ABC}\cong △\mathrm{DEF}?$