Use the given figure to prove that, AB=AC.

Question

Use the given figure to prove that, $\text{AB}=\text{AC}$.

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Answer

In $\triangle APQ,$
$\text{AP} = \text{AQ} .........[$Given$]$
$\therefore \angle APQ = \angle AQP .........(i) [$angles opposite to equal sides are equal$]$
In $\triangle ABP,$
$\angle APQ = \angle BAP + \angle ABP .......(ii) [$Ext. the angle is equal to the sum of opp. int. angles$]$
In $\triangle AQC,$
$\angle AQP = \angle CAQ + \angle ACQ ...(iii) [$Ext. angle is equal to sum of opp. int. angles$]$
From $(i), (ii)$ and $(iii)$
$\angle BAP + \angle ABP = \angle CAQ + \angle ACQ$
But, $\angle BAP = \angle CAQ .......[$Given$]$
$\Rightarrow \angle CAQ + \angle ABP = \angle CAQ + \angle ACQ$
$\Rightarrow \angle ABP = \angle CAQ + \angle ACQ − \angle CAQ$
$\Rightarrow \angle ABP = \angle ACQ$
$\Rightarrow \angle B = \angle C ...........(iv)$
In $\triangle ABC,$
$\angle B = \angle C$
$\Rightarrow \text{AB} = \text{AC} ......[$Sides opposite to equal angles are equal$]$

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