Question

Q.30 In the fig.10, D and E are points on the base BC of a AABC such that AD = AE
and BAD = ZCAE. Prove that AB = AC.​

Answer 1

Answer:

Given:- D and E are points on the base BC of a ∆ABC such that AD = AE and ∠BAD = ∠CAE.

To Prove:- AB = AC

Proof:- In ∆ADE,

∵ AD = AE    (Given )

∴ ∠ADE = ∠AED    ....(1)

| Angles opposite to equal sides of a triangle are equal

In ∆ABD,

Ext. ∠ADE = ∠BAD + ∠ABD ....(2)

| An exterior angle of a triangle is equal to the sum of its two interior opposite angles

In ∆AEC,

Ext. ∠AED = ∠CAE + ∠ACE ....(3)

| An exterior angle of a triangle is equal to the sum of its two interior opposite angles

From (1), (2) and (3),

∠BAD + ∠ABD = ∠CAE + ∠ACE

⇒ ∠ABD = ∠ACE

| ∵ ∠BAD = ∠CAE (Given)

⇒ ∠ABC = ∠ACB

∴ AB = AC

| Sides opposite to equal angles of a triangle are equal