CAT Angle Bisector Theorem - Practice Questions & MCQ

Theorem: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Let us prove this.

Given: In ΔABC, AD is the bisector of ∠A, and the bisector of ∠A meets BC in D.

We need to prove: .

Construction: Draw CP parallel to AD to meet BA produced at P.

 Proof:

∠DAC = ∠ACP                  (alternate angles and AD || CP)

∠BAD = ∠APC                  (corresponding angles)

But ∠BAD = ∠D                (given)

∴ ∠ACP = ∠APC

In triangle APC,

AC = AP                             (sides opposite to equal angles are equal) In triangle BCP,

                    (by basic proportionality theorem)

Hence Proved.

The converse of Angle Bisector Theorem

If a line that passes through a vertex of a triangle, divides the base in the ratio of the other two sides, then it bisects the angle.

In the figure, AD divides BC in the ratio  and if  then AD is the bisector of ∠A.