Question

∆ABC is an isosceles triangle in which AB = AC. Also, D is a point such that BD = CD. Prove that AD bisects angle A and angle D.

Answer 1

Step-by-step explanation:

here your answer mate . hope it helps you

Answer 2

Given,

∆ABC is an = isosceles triangle

AB = AC

And, BD = CD

To find,

The AD bisects angle A and angle D.

Solution,

We have to apply the congruent properties of the triangles to solve this mathematical problem.

In ∆ABO and ∆ACO

AB = AC (Given in the question.)

AO is the common side.

∠ABO = ∠ACO

(Because, two base angles of a isosceles triangle are equal and ∠ABO and ∠ACO are two base angles of the isosceles triangle ∆ABC)

So,by the SAS (Side-Angle-Side) property, we can say that ∆ABO and ∆ACO are congruent.

So, ∠BAO = ∠CAO ......(1)

And, BO = CO

Now, in ∆BOD and ∆COD

BO = CO

OD = Common side

BD = CD (given in question)

So, by SSS (side-side-side) property the ∆BOD and ∆COD are congruent triangles.

So, ∠BDO = ∠CDO.....(1)

By, (1) and (2) we can say that AD bisects ∠A and ∠D

Hence, AD is the bisector of ∠A and ∠D.