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Question
On the sides AB and AC of triangle ABC, equilateral triangle ABD and ACE are drawn. Prove that:
- ∠CAD = ∠BAE
- CD = BE
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Solution
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.
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