# In Fig. 6.40, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC,prove that Δ ABD ~ Δ ECF

To Prove: Δ ABD ∼ Δ ECF

Given: ABC is an isosceles triangle, AD is perpendicular to BC

BC is produced to E and EF is perpendicular to AC

Proof:

Given that ABC is an isosceles triangle

AB = AC

⇒ ∠ABD = ECF

In ΔABD and ΔECF,

ABD = ∠ECF (Proved above)

Therefore,

ΔABD ΔECF (By using AA similarity criterion)

AA Criterion: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Hence, Proved.

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