Chapter 8.3, Problem 14PSB

To determine

To prove : The similarity of $\Delta ABC$ with respect to $\Delta DEF$ .

Explanation of Solution

Given information : Here, we have given that vertex angle of one isosceles triangle is similar to the vertex angle of another triangle. $\Rightarrow \angle ABC\cong \angle DEF$ )

  

Proof: As, $\Delta ABC$ and $\Delta DEF$ are isosceles triangles,

  $\Rightarrow \overline{AB}=\overline{AC}$ and $\overline{DE}=\overline{DF}$

  $\Rightarrow \angle ABC=\angle ACB$ and $\angle DEF=\angle DFE$ (Angles opposite to equal sides are equal)

Now, as $\angle ABC\cong \angle DEF$ ,

  $\Rightarrow \angle ABC=\angle ACB$ and $\angle ABC=\angle DFE$

  $\Rightarrow \angle ACB=\angle DFE$

Now, In $\Delta ABC$ and $\Delta DEF$

  $\angle ABC\cong \angle DEF$ (Given)

  $\angle ACB\cong \angle DFE$ (Proved above)

  $\Rightarrow \Delta ABC\sim \Delta DEF$ (By AA Similarity Rule)