Name: Angle Sum Property of a Triangle
Uploaded: 2019-07-19
Duration: 3 min 14 s
Description: No matter what kind of triangle is analysed, you will always find that the sum of its angles is 180 degrees. Watch and learn the mathematics behind this in d...

Question

A B C

is a triangle. The bisector of the exterior angle at

B

and the bisector of

∠ C

intersect each other at

D .

Prove that

∠ D = \frac{1}{2} ∠ A .

Accepted Answer

In the given $△ A B C$ , the bisectors of $e x t . ∠ B$ and $∠ C$ intersect at $D$ .

We need to prove: $∠ D = \frac{1}{2} ∠ A$ .

Now, using the exterior angle theorem,

$∠ A B E = ∠ B A C + ∠ A C B$

As $∠ A B E$ and $∠ A C B$ are bisected

$∠ D C B = \frac{1}{2} ∠ A C B$ .

Also, $∠ D B A = \frac{1}{2} ∠ A B E$ .

Further, applying angle sum property of the triangle

In $△ D C B$ ,

$∠ C D B + ∠ D C B + ∠ C B D = 180 °$

$\Rightarrow ∠ C D B + \frac{1}{2} ∠ A C B + (∠ D B A + ∠ A B C) = 180 °$

$∠ C D B + \frac{1}{2} ∠ A C B + (\frac{1}{2} ∠ A B E + ∠ A B C) = 180 °$ $\dots \dots (2)$

Also, $C B E$ is a straight line, So, using linear pair property

$\Rightarrow ∠ A B C + ∠ A B E = 180 °$

$\Rightarrow ∠ A B C + \frac{1}{2} ∠ A B E + \frac{1}{2} ∠ A B E = 180 °$

$\Rightarrow ∠ A B C + \frac{1}{2} ∠ A B E = 180 ° - \frac{1}{2} ∠ A B E$ $\dots \dots (3)$

So, using $(3)$ in $(2)$

$∠ C D B + \frac{1}{2} ∠ A C B + (180 ° - \frac{1}{2} ∠ A B E) = 180 °$

$\Rightarrow ∠ C D B + \frac{1}{2} ∠ A C B - \frac{1}{2} ∠ A B E = 0$

$\Rightarrow ∠ C D B = \frac{1}{2} (∠ A B E - ∠ A C B)$

$\Rightarrow ∠ C D B = \frac{1}{2} ∠ C A B$

$\Rightarrow ∠ D = \frac{1}{2} ∠ A$

Hence proved.

$A B C$ is a triangle. The bisector of the exterior angle at $B$ and the bisector of $∠ C$ intersect each other at $D .$ Prove that $∠ D = \frac{1}{2} ∠ A .$

Important Questions on Triangle and its Angles

Learn and Prepare for any exam you want !

ABC is a triangle. The bisector of the exterior angle at B and the bisector of ∠C intersect each other at D. Prove that ∠D=12∠A.

Important Questions on Triangle and its Angles

Learn and Prepare for any exam you want !

$A B C$ is a triangle. The bisector of the exterior angle at $B$ and the bisector of $∠ C$ intersect each other at $D .$ Prove that $∠ D = \frac{1}{2} ∠ A .$