Chapter 3, Problem 18RP

To determine

To Prove: $\Delta DFE$ is isosceles

Explanation of Solution

Given information: $\overline{AC}\cong \overline{BC}$ and $\angle 1\cong \angle 3$ is provided in the question which will help to find the solution.

Proof: Here, $\overline{AC}\cong \overline{BC}$ (given)

And, $\angle CAB=\angle CBA$ (angle opposite to equal sides are equal)

  $\angle 1\cong \angle 3$ (given)

And, EC = CD (sides opposite to equal angles are equal)

Here, $\angle CDF=\angle CEF$ (right angle)

So that, $\angle 1+\angle EDF=\angle 3+\angle DEF$ ( $\because \angle 1=\angle 3$ )

So, $\angle EDF=\angle DEF$

Hence, DF = EF (sides opposite to equal angles are equal)

Hence, we can say that

  $\Delta DFE$ is isosceles.