Can two equilateral triangles always be congruent? Give reasons.

In the given figure,$PA\perp AB, QB\perp AB \mathrm{and} PA=QB$. Prove that $∆OAP\cong ∆OBQ$. Is $OA=OB$?

In the given figure, $∆ABC$ is isosceles in which $AB=AC$. If $E$ and $F$ are the midpoints of $AC$ and $AB$ respectively. Prove that $BE=CF$.

In the given figure, $AB=AC \mathrm{and} BD=DC$. Prove $∆ADB \cong ∆ADC$ and hence show that $\angle ADB= \angle ADC=90°$ and $\angle BAD=\angle CAD$.

In the given figure, $∆ABC$ is isosceles in which $AB=AC$. If $AB$ and $AC$ are produced to $D$ and $E$ respectively such that $BD=CE$. Prove that $BE=CD$.

In the given figure, triangles $ABC \mathrm{and} DCB$ are right-angled at $A$ and $D$ respectively and $AC=DB$. Prove that $∆ABC\cong ∆DCB$.

In the given figure, $P$ and $Q$ are two points on equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ such that $AP=AQ$. Prove that $BQ=CP$.

In the given figure, $ABC$ is a triangle in which $AD$ is the bisector of $\angle A$. If $AD\perp BC$, show that $∆ABC$ is isosceles.

In the given figure, $∆ABC$ is an isosceles triangle in which $AB=AC$. Also, $D$ is a point such that $BD=CD$. Prove that $AD$ bisects $\angle A$ and $\angle D$.

In the given figure, $AD=BC$ and $AD\left|\right|BC$. Is $AB=DC$? Give reasons in support of your answer.

In the given figure, $AB=AD \mathrm{and} CB=CD$. Prove that $∆ABC\cong ∆ADC$.