$P$ is a point on the side BC of $△ABC$ such that $PL\perp AB \text{and} PM\perp AC \text{and} PL=PM$ . Show that $AP$ bisects $\angle BAC.$

In $△ABC and △DEF, AL\perp BC, DM\perp EF, AB =DE, AC=DF,AL=DM.$ . Show that $△ABC \cong △DEF.$

$ABCD$ is a square. $P and Q$ are points on $DC\text{ and} BC$ respectively, such that $AP=DQ$ . Prove that $∆ADP\cong ∆DCQ$

In the given figure, $\mathrm{DA}\perp \mathrm{CD}$ and $\mathrm{CB}\perp \mathrm{CD}$. If $\mathrm{DP}=\mathrm{CQ}$ and $\mathrm{AQ}=\mathrm{BP}$. Prove that $\angle \mathrm{DAQ}=\angle \mathrm{CBP}$

Two equal chords $\mathrm{AB}$ and $\mathrm{CD}$ are drawn in a circle with centre $O$. Show that $∆\mathrm{OAB}$ and $∆\mathrm{OCD}$ are congruent.

$\mathrm{ABCD}$ is a square. $A$ is joined to $X$ on $BC$ and $D$ to a point $Y$ on $AB$ as shown in the figure. If $AX=BY$, prove that $AX$ and $DY$ are at right angles.