In the following diagram, $ABCD$ is a square and $APB$ is an equilateral triangle.

Prove that $△APD=△BPC$.

In the following diagrams, $ABCD$ is a square and $APB$ is an equilateral triangle.

Find the angles of $△DPC$.

In the figure, given below, triangle $ABC$ is right-angled at $B. ABPQ$ and $ACRS$ are squares. Prove that $△ACQ$ and $△ASB$ are congruent.

In the figure, given below, triangle $ABC$ is right-angled at $B. ABPQ$ and $ACRS$ are squares. Prove that $CQ=BS$.

In the adjoining figure, $QX$ and $RX$ are the bisectors of the angles $Q$ and $R$ respectively of the triangle $PQR$. If $XS\perp QR$ and $XT\perp PQ$; prove that $△XTQ\cong △XSQ$.

In the adjoining figure, $QX$ and $RX$ are the bisectors of the angles $Q$ and $R$ respectively of the triangle $PQR$. If $XS\perp QR$ and $XT\perp PQ$; prove that $PX$ bisects angle $P$.

In a parallelogram $ABCD$, the angles $A$ and $C$ are obtuse. Points $X$ and $Y$ are taken on the diagonal $BD$ such that $\angle XAD$ and $\angle YCB$ are right angles. Prove that $XA=YC$.