Name: AAS Criteria for Congruence of Triangles: Example
Uploaded: 2019-07-19
Duration: 2 min 58 s
Description: When two angles and a non-included side of a triangle are equal to the corresponding angles and sides of another triangle, then two triangles are congruent. Wat

Question

If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

Accepted Answer

Let the triangle be $∆ ABC$ and $AD$ be the angle bisector of $∠ BAC$ .
Also $BD = DC$ .

To prove the $∆ ABC$ is isosceles.

Produce $AD$ up to $E$ such that $AD = DE$ .

In $∆ ABD$ and $∆ EDC,$

$\Rightarrow AD = DE$ (by construction)

$\Rightarrow BD = CD$ (given)

$\Rightarrow ∠ 1 = ∠ 2$ (vertically opposite angles)

By using $S . A . S$ congruency rule,

$∴ ∆ ABD = ∆ EDC$

Corresponding parts of congruent triangles are equal.
$\Rightarrow AB = CE . . . . . . . . (1)$

And $∠ BAD = ∠ CED$ .

Given, $AD$ is bisector of $∠ BAC$ .

$∵ ∠ BAD = ∠ CAD$

$\Rightarrow ∠ CED = ∠ CAD$

We know that, sides opposite to equal angles are equal.

$\Rightarrow AC = CE . . . . . . . . (2)$

From $(1)$ and $(2)$ ,

$∴ AB = AC$ .

A triangle with two equal sides is known as an isosceles triangle.

Thus, $∆ ABC$ is an isosceles triangle.

Hence, proved.

If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

Important Questions on Triangles

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