Chapter 3.7, Problem 22PSC

To determine

To prove : $\Delta FJK$ is isosceles.

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{FG}\cong \overrightarrow{JH}$

  $\angle FGH\cong \angle JHG$

Formula used : Similar triangles have congruent angles with the original triangle

Proof : In triangles FGH and JHG, we can say that

  $\overrightarrow{FG}\cong \overrightarrow{JH}$

  $\angle FGH\cong \angle JHG$

  $\overrightarrow{GH}\cong \overrightarrow{HG}$ (common side)

Thus, by SAS property, we can say that the triangles FGH and JHG are congruent

Now, if the triangles FGH and JHG are congruent, then we can say that

  $\Delta FKG\cong \Delta JKH$ (By similar triangles property)

Thus, if $\Delta FKG\cong \Delta JKH$ , then we can say that

  $\overrightarrow{FK}\cong \overrightarrow{JK}$

Now, in $\Delta FJK$ if $\overrightarrow{FK}\cong \overrightarrow{JK}$ , then we can say that

  $\Delta FJK$ is isosceles

Hence, required details are proved