Question

$ABCD$ is a square. $A$ is joined to a point on $BC$ and $D$ is joined to a point $Q$ on $AB$. If $AP=DQ$; prove that $AP$ and $DQ$ are perpendicular to each other.

Answer 1

Given, $DQ=AD$

$\Rightarrow ABCD$ is a square

In $△APB$ and $△AQD$

$AD=DQ$ (given)

$AB=AD$ (side of square)

By, R.H.S rule, $△ABP=△ADR$

$x=\angle ADQ=\angle PAB$ and $\angle APB=\angle AQD=y$

$x+y={90}^o$

In $△APQ,\angle ARQ={90}^o$

$=180(x+y)=180-90=90$

$AP$ is perpendicular to $DQ$