Question

In the given figure, $AB=AD,\angle 1=\angle 2=\angle 3=\angle 4$. Prove that $AP=AQ$.

Answer 1

Given is a figure where $AB=AD,\angle 1=\angle 2=\angle 3=\angle 4$

To prove: $AP=PQ$

Proof:-

$AD=AB$ (Given)

$\angle 3=\angle 4$ (Given)

$\angle 1=\angle 2$ (Given)

$\therefore \angle 3+\angle 1=\angle 4+\angle 2$

Hence, $\angle BAC=\angle DAC$

$AC=AC$ [common]

$\therefore \Delta ABC\cong \Delta ADC$ (SAS rule)

Hence, $\angle BCA=\angle DCA\left(CPCT\right)$

Now, in $\Delta APCand\Delta AQC$,

$\angle 3=\angle 4$ (Given)

$AC=AC$ [common]

and $\angle PCA=\angle QCA[\because \angle BCA=\angle DCA]$

Hence, $\Delta APC\cong \Delta AQC$ [ASA rule]

$\therefore AP=AQ\left[CPCT\right].\left[Henceproved\right]$