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# In quadrilateral $A C B D$,$\mathrm{AC}=\mathrm{AD}$ and $\mathrm{AB}$ bisects $\angle \mathrm{A}$ (see Fig. 7.16). Show that $\triangle \mathrm{ABC} \cong \triangle \mathrm{ABD}$.What can you say about $\mathrm{BC}$ and $\mathrm{BD}$ ?"64391"

Given:

In quadrilateral $ABCD, AC=AD$ and bisects $\angle A$.

To do:

We have to show that $\triangle ABC\cong ABCD$ and say about $BC$ and $BD$.

Solution:

Let us consider $\triangle ABC$ and $\triangle ABD$.

Given,

$AC=AD$

The line segment $AB$ bisects $\angle A$.

Therefore,

$\angle CAB=\angle DAB$

We know that,

According to Rule of Side-Angle-Side Congruence:

Triangles are said to be congruent if any pair of corresponding sides and their included angles are equal in both triangles.

Therefore,

$\triangle ABC\cong ABCD$

We also know that,

From corresponding parts of congruent triangles: If two triangles are congruent, all of their corresponding angles and sides must be equal.

This implies,

$BC=BD$.

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Updated on: 10-Oct-2022

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