Question
In the given figure, $AD$ and $BC$ are equal perpendiculars to a line segment $AB$. Show that $CD$ bisects $AB.$ 
✓
Answer
In $\triangle\text{AOD}$ and $\triangle\text{BOC},$
$\angle\text{AOD}=\angle\text{BOC}$ (vertically opposite angle)
$\angle\text{DAO}=\angle\text{CBO} ($Each $90^\circ )$ $\text{AD = BC}$ (given)
$\therefore\triangle\text{AOD}\cong\triangle\text{BOC} ($by $AAS$ congruence criterion$)$
$\Rightarrow\text{AO = BO} (C.P.C.T.)$
$\Rightarrow\text{CD bisects AB}.$
$\angle\text{AOD}=\angle\text{BOC}$ (vertically opposite angle)
$\angle\text{DAO}=\angle\text{CBO} ($Each $90^\circ )$ $\text{AD = BC}$ (given)
$\therefore\triangle\text{AOD}\cong\triangle\text{BOC} ($by $AAS$ congruence criterion$)$
$\Rightarrow\text{AO = BO} (C.P.C.T.)$
$\Rightarrow\text{CD bisects AB}.$
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