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Two lines $AB$ and $CD$ intersect at $O$ such that $BC$ is equal and parallel to $AD$. Prove that the lines $AB$ and $CD$ bisect at $O$.
Given:
Two lines $AB$ and $CD$ intersect at $O$ such that $BC$ is equal and parallel to $AD$.
To do:
We have to prove that the lines $AB$ and $CD$ bisect at $O$.
Solution:
$BC = AD$ and $BC \parallel AD$
In $\triangle AOD$ and $\triangle BOC$,
$AD = BC$
$\angle A = \angle B$ (Alternate angles are equal)
$\angle D = \angle C$ (Alternate angles)
Therefore, by ASA axiom,
$\triangle AOD \cong \triangle BOC$
This implies,
$AO = OB$ (CPCT)
$AO = OC$ (CPCT)
Hence, $AB$ and $CD$ bisect each other at $O$.
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