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Equal Angles Subtend Equal Sides

In ABC, AB=...

Question

In $△ A B C, A B = A C$ . $B C$ is a extended on both sides to $E$ and $D$ respectively such that $V E = B D$ . Prove that $A D = A E$ .

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Solution

$\begin{matrix} A B = A C (i s o s c e l e s t r i a n g l e A B C) \to (i) E C = B D (g i v e n) \Rightarrow E B + B C = B C + C D \Rightarrow E B = C D \to (i i) ∠ A B C = ∠ A C B (i s o s c e l e s t r i a n g l e A B C) 180 - ∠ A B C = 180 = ∠ A C B \Rightarrow ∠ A B E = A C D \to (i i i) f r o m e q u a t i o n (i), (i i), (i i i) Δ A B E ≅ Δ A C D b y C P C T A E = A D h e n c e p r o v e d . \end{matrix}$

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