Question

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

Prove that: (i) $\angle$ CAD = $\angle$ BAE

(ii) CD = BE.

Answer 1

Given: ABC is a triangle. ABD and ACE are equilateral

RTP: CD = BE

Construction: Join BE and CD.
Proof:
$\angle BAD$= $\angle CAE$ [Angles of equilateral triangle]

$\angle BAD$ + $\angle BAD$ = $\angle CAE$ + $\angle BAC$

[By adding $\angle BAC$ on both sides]


In $△ACD$ and $△ABE$
AD = AB
AC = AE [equilateral triangle]

$\angle CAD$= $\angle BAE$ {Proved}

$△ACD$ = $△AEB$ [According to SAS congruency]

Therefore CD = BE.