Question

On the sides AB and AC of triangle ABC, equilateral triangle ABD and ACE are drawn. Prove that: 

1. ∠CAD = ∠BAE
2. CD = BE

Answer 1

Given:  ΔABD is an equilateral triangle.

ΔACE is an equilateral triangle

We need to prove that 

(i) ∠CAD = ∠BAE 

Proof:
(i) ΔABD is equilateral

∴ Each angle = 60°

⇒ ∠BAD = 60°                            ...(1)

Similarly,

 ΔACE is equilateral

∴ Each angle = 60°

⇒ ∠CAE = 60°                             ...(2)

⇒ ∠BAD = ∠CAE      ...[ from (1) and (2) ]...(3)

Adding ∠BAC to both sides, we have

⇒ ∠BAD + ∠BAC = ∠CAE + ∠BAC

⇒ ∠CAD = ∠BAE                         ...(4)

(ii) In ΔCAD and ΔBAE

AC = AE                 ...[ ΔACE is equilateral]

∠CAD = ∠BAE      ... [ from (4) ]

AD = AB                ...[ ΔABD is equilateral]

∴ By the Side-Angle-Side criterion of congruency, 

ΔCAD ≅ ΔBAE

The corresponding parts of the congruent

triangles are congruent.

∴ CD = BE             ...[ by c.p.c.t ]

Hence proved.