In the following figure, AE = AD, $∠ B A E = ∠ C A D$ . Prove that AB = AC.

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Solution

Since AE = AD, so $∠ A E D = ∠ A D E$ . Now
$180^{\circ} - ∠ A E D = 180^{\circ} - ∠ A D E \Rightarrow ∠ A E B = ∠ A D C$
In $∆ A B E and ∆ A C D$
$A E = A D$ (Given)
$∠ B A E = ∠ C A D$ (Given)
$∠ A E B = ∠ A D C$ (Prove above)
$∴ ∆ A B E ≅ ∆ A C D$ (By ASA)
Thus, by corresponding parts of congruent triangles
AB = AC

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