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Triangles — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsTriangles3 Marks
Question
In the figure, $E$ is the point on side $CB$ produced on an isosceles triangle $ABC$ with $AB = AC$. If $AD$ $ \bot $ $BC$ and EF$ \bot $ $AC,$ prove that $\triangle $ABD $ \sim $ $\triangle $ECF.
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Answer

$E$ is the point on side $C B$ produced on an isosceles triangle $A B C$ with $A B=A C \cdot A D-B C \perp$ and $E F \perp A C$. with $A B=A C$. Also, $A D \perp B C$ and $E F \perp A C$.
To prove: $\triangle ABD \sim \triangle ECF$
Proof: In $\triangle A B D$ and $\triangle E C F$,
$\therefore A B=A C$........Given
$\therefore \angle A C B=\angle A B C$......Angle opposite to equal sides of a triangle are equal
$\Rightarrow \angle A B C=\angle A C B$
$\Rightarrow \angle A B D=\angle E C F$.........(1)
$\angle A D B=\angle E F C$.
(2) [Each equal to $90^{\circ} In$ view of (1) and (2)]
$\triangle ABD \sim \triangle ECF$
..AA similarity criterion

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