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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Updated on: 10-Oct-2022

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