Question

Two line segments AB and CD bisect each other at O. Prove that
i) AC = BD
ii) AD || CB
iii) $\angle CAB=\angle ABD$
iv) AD = CB

Answer 1

AB and CD bisect each other at O i.e, $AO=BO$ and $CO=DO$

in $\Delta COA$ and $\Delta DOB$

$CO=OD$ [Given]

$\angle COA=\angle BOD$ [ vertically opp angles]

$AO=BO$

$\therefore \Delta COA\cong \Delta BOD$ by SAS rule.

(i) $\therefore AC=BD[C.P.CT]$

(iii) $\angle CAB=\angle ABD[C.P.CT]$

Now,

In $\Delta COB$ and $\Delta AOD$

$CO=OD$ [given]

$BO=AO$ [given]

$\angle COB=\angle AOD$ [vertically opp angles]

$\therefore \Delta COB\cong \Delta AOD$ by SAS rule.

$\therefore \angle CBA=\angle BAD$ [ C.P. C.T]

(ii) and so $AD||CB$ [ $\because \angle CBA=\angle BAD$ are a pair of alternate angles]

(iv)and $AD=CB$ [C.P. C.T]