Chapter 3, Problem 6RP

To determine

To prove : $\Delta HCE$ is an isosceles triangle.

Explanation of Solution

Given information : Here, we have given that $\overline{DG}\cong \overline{JF}$ , $\overline{DE}\cong \overline{JH}$ and $\overline{EG}\cong \overline{HF}$ .

Proof: In $\Delta HJF$ and $\Delta EDG$ ,

  $\overline{DG}\cong \overline{JF}$ (Given)

  $\overline{DE}\cong \overline{JH}$ (Given)

  $\overline{EG}\cong \overline{HF}$ (Given)

  $\Rightarrow \Delta HJF\cong \Delta EDG$ (By SSS Congruency Rule)

  $\Rightarrow \Delta HJF\cong \Delta EDG$ (By Corresponding Part of Congruent Triangles)

Now, as opposite angles of $\Delta HCE$ are equal

  $\Rightarrow \overline{CE}\cong \overline{CH}$ (Sides opposite to equal angles are equal in a triangle)

Now, as two sides of $\Delta HCE$ are equal,

  $\therefore$

  $\Delta HCE$ is an isosceles triangle.