Ex.6.3 Q11 Triangles Solution - NCERT Maths Class 10

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Question

In Figure \(, E\) is a point on side \(CB\) produced of an isosceles \(\triangle ABC\) with \(AB = AC\). If \(AD \bot BC\) and \(EF \bot AC\), prove that \(\Delta A B D \sim \Delta E C F\).

Diagram

 

Text Solution

  

Reasoning:

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

This is referred as the \(AA\) criterion for two triangles.

Steps:

In \(\Delta ABD,\,\,\Delta ECF \)

\(\begin{align} \angle ADB&=\angle EFC={{90}^{\circ }}\qquad (\because AD\bot BC \text{and}\,EF\bot AC) \\ \angle ABD& =\angle ECF \qquad (\because \text{In}\,\Delta ABC,AB=AC \Rightarrow \angle ABC=\angle ACB) \\ \Rightarrow &\Delta ABD \sim \Delta ECF\;\; \qquad(\text{AA}\,\,\text{criterion}) \\ \end{align}\)