Line $ l $ is the bisector of an angle $ A $ and $ B $ is any point on $l$. BP and BQ are perpendiculars from B to the arms of $ \angle A $. Show that
(i) $ \triangle \mathrm{APB} \cong \triangle \mathrm{AQB} $
(ii) $ \mathrm{BP}=\mathrm{BQ} $ or $ \mathrm{B} $ is equidistant from the arm of angle $ \mathrm{A} $
Given:
\( l \) is the bisector of \( \angle A \) and $B$ is a point on $l$.
BP and BQ are perpendiculars from $B$ to the arms of $\angle A$.
To do:
\( \angle P A B=\angle Q A B \).......(i)
\( \angle \mathrm{APB}=\angle \mathrm{AQB}=90^{\circ} \)...........(ii)
In \( \triangle \mathrm{APB} \) and \( \triangle \mathrm{AQB} \),
$\angle P A B=\angle Q A B$
$\angle \mathrm{APB}=\angle \mathrm{AQB}$
$A B=A B$ (common side)
Therefore, by AAS congruency,
$\triangle \mathrm{APB} \cong \triangle \mathrm{AQB}$
This implies,
$\mathrm{BP}=\mathrm{BQ}$ (CPCT)
Hence proved.
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