Question

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

Answer 1

Answer:

The proof is explained step-wise below :

Step-by-step explanation:

For better understanding of the solution, see the attached figure :

In ΔABD and ΔEDC

AD = DE ( By construction)

∠1 = ∠2 ( Vertical opposite angles are equal)

BD = CD ( Given )

By SAS congruency criterion , ΔABD ≅ ΔEDC

So, By using the property of congruency , Corresponding parts of congruent triangles are equal

⇒ AB = CE............(1) and ∠BAD = ∠CED

Also, ∠BAD = ∠CAD ( AD is bisector of ∠BAC)

⇒ ∠CED = ∠CAD

⇒ AC = CE .......(2)

From equation (1) and equation (2)

AB = AC

So, ABC is an isosceles triangle.

Hence Proved.