Chapter 8, Problem 21RP

To determine

To prove:The given two triangles are similar.

Explanation of Solution

Given information:

ABDF is a parallelogram

Formula used:

Alternate Interior angles are equal

AAA criteria of similarity

Proof:

In triangles DEF and BCD, we have

  $\begin{array}{l}\angle \text{CBD}=\angle \text{BDF}\mathrm{.....}\text{ interior angles}\\ \therefore \angle \text{DFE}=\angle \text{BDF}\mathrm{.....}\text{interior angles}\\ \text{DF}\text{}\text{= BC}\\ \angle \text{EDF}=\angle \text{DCE}\mathrm{.....}\text{(a)}\\ \angle \text{CDB}=\angle \text{DEF}\mathrm{.....}\text{(b)}\\ \angle \text{DBC}=\angle \text{DFF}\mathrm{.....}\text{(c)}\end{array}$

From 9a), (b) and (C), we have

  $\Delta DEF\sim \Delta BCD$ ….. AAA criteria of similarity satisfied

Hence proved