Chapter 3.7, Problem 19PSB

To determine

To prove : $\Delta JOK$ is isosceles.

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{HJ}\cong \overrightarrow{MK}$

  $\angle HJK\cong \angle MKJ$

Formula used : Similar sides of congruent triangles are congruent

Proof : We compare the triangles HOJ and MOK. Thus, we get

  $\overrightarrow{HJ}\cong \overrightarrow{MK}$

  $\angle HOJ\cong \angle MOK$ (vertically opposite angles)

  $\angle JHO\cong \angle KMO$

Hence, we can say that the triangles $\Delta HOJ\cong \Delta MOK$ by ASA property

Now, as we know that the similar sides of congruent triangles are congruent, hence we can say that

  $\overrightarrow{JO}\cong \overrightarrow{KO}$

Now, in $\Delta JOK$ , we have two congruent sides

  $\overrightarrow{JO}\cong \overrightarrow{KO}$

Thus, any triangle with two congruent sides is an isosceles triangle.