Question

To prove the theorem of an angle bisector
of a triangle do the following activity
(i) Draw a neat labelled figure
( 2)write : Given and to prove​

Answer 1

$\Large\bf\underline{Angle\:Bisector\:Theorem}$

Bisector of an angle divides opposite sides in the same ratio formed by its arms.

Given:

In $\triangle$ABC

AD bisects $\angle$A.

To prove:

$\frac{BD}{DC} = \frac{AB}{AC}$

Construction:

Construct CE || DA

Proof:

In $\triangle$EBC

AD || EC

$\implies \angle3 = \angle4${Alternate Angles}----(i)

$\implies \angle1 = \angle2${Corresponding Angles}(ii)

AD bisect $\angleA$

$\implies \angle1 = \angle3$----(iii)

From (i),(ii) and (iii)

$\implies \angle2 = \angle4$

$\implies$AE=AC----(iv)

In $\triangle$EBC,

CE || AD

$\implies \frac{AB}{AE} = \frac{BD}{DC}$

From (iv)

$\implies \frac{AB}{AC} = \frac{BD}{DC}$

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