Chapter 3.7, Problem 21PSB

To determine

To prove : $\Delta CDG$ is isosceles.

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{CE}\cong \overrightarrow{CF}$

  $\angle F\cong \angle 3$

  $\angle E$ is supplementary to $\angle 5$

Formula used : Lines parallel to the base of a triangle will make similar triangles to the original one

Proof : In triangle CEF, we know that $\overrightarrow{CE}\cong \overrightarrow{CF}$ , hence, we can say that

  $\Delta CEF$ is isosceles

Now, we know that

  $\angle F\cong \angle 3$

  $\angle E$ is supplementary to $\angle 5$

Thus, we can say that the line DG is parallel to EF

  $\overrightarrow{DG}\left|\right|\overrightarrow{EF}$

Now, we know that lines parallel to the base of a triangle will make similar triangles to the original one

Thus, we can say that triangle CDG is similar to CEF. Hence, if $\Delta CEF$ is isosceles, then

  $\Delta CDG$ is isosceles