Monica Capoor Solutions for Exercise 7: Exercise 13.5

In the pair of right triangles, the measures of some parts are indicated. By applying the RHS congruence rule, state the pair of triangles are congruent or not. In case of congruent triangles, write the result in symbolic form.

$ABC$ is an isosceles triangle in which $AB=AC$. Using the RHS property, prove that $\angle B=\angle C$.

In figure, line segments $AB \mathrm{and} CD$ bisect each other at $O$. $AC\perp CD \mathrm{and} BD\perp CD$. Prove that $△ACO\cong △BDO$ and hence prove that $AC=BD$.

In figure, $△BAC \mathrm{and} △CDB$ are right triangles, right-angled at $A \mathrm{and} D$ respectively. If $AC=DB$, prove that $△BAC\cong △CDB$.

In figure, $AD \mathrm{and} BE$ are respectively the altitudes of $△ABC$ such that $AE=BD$. Prove that $AD=BE$.

In figure, $D$ is the midpoint of $BC$. $DP\perp AB, DQ\perp AC \mathrm{and} DP=DQ$. Prove that $\angle B=\angle C$.

In figure, $P$ is any point on $BX$. $MPL$ is perpendicular to $BX$ and it meets $BC$ at $M$ and $AB$ at $L$ such that $BM=BL$. Prove that $PM=PL$.

$ABCD$ is a rectangle. $X \mathrm{and} Y$ are points on side $AD \mathrm{and} BC$ respectively such that $AY=BX$. Show that $BY=AX \mathrm{and} \angle BAY=\angle ABX$.