Ex.6.3 Q6 Triangles Solution - NCERT Maths Class 10

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Question

In Figure , if \(\Delta \mathrm{ABE} \cong \Delta \mathrm{ACD}\), show that \(\Delta ADE \sim \Delta ABC\).

Diagram

Text Solution

 

Reasoning:

As we know if two triangles are congruent to each other;their corresponding parts are equal.

If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

This criterion is referred to as the SAS (Side–Angle–Side) similarity criterion for two triangles.

Steps:

In \(\begin{align}\Delta ABE\,\,{\rm {and}}\,\Delta ACD \end{align}\)

\[\begin{align}& AE=AD\,\,(\because \Delta ABE\cong \Delta ACD\,\,\rm{given}).........(1) \\ & AB=AC\,\,(\because\Delta ABE\cong \Delta ACD\,\,\rm{given}).........(2) \\ \end{align}\]

Now Consider \(\Delta ADE,\,\,\Delta ABC\)

\[\begin{align} \frac{AD}{AB}&=\frac{AE}{AC}\qquad\,\text{from}\,\,(1)\,\And (2) \\  \rm{and}\quad\,\,\angle DAF&=\angle BAC\,\,\,\left( \text{Common}\,\text{angle} \right) \\  \Rightarrow\quad \Delta ADE&\sim{\ }\Delta ABC\left( \text{SAS}\,\text{criterion} \right) \\ \end{align}\]