Chapter 3.2, Problem 8PSA

To determine

To prove : The congruency of $\Delta BDG\cong \Delta FDG$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{BC}\cong \overrightarrow{FE}$

  $\overrightarrow{DC}\cong \overrightarrow{DE}$

  $\angle 5\cong \angle 6$

Formula used : Two triangles are congruent by Side-angle-side property, if there are two congruent sides, and the angle between them is also congruent

Proof : We know that

  $\overrightarrow{BC}\cong \overrightarrow{FE}$

  $\overrightarrow{DC}\cong \overrightarrow{DE}$

Now, adding up both of these equations, we get

  $\overrightarrow{DB}\cong \overrightarrow{DF}$

Also, we know that

  $\angle 5\cong \angle 6$

And, as we can see, the chord DG is a common chord

Now, by SAS property mentioned above, we can say that

  $\Delta BDG\cong \Delta FDG$

Hence, required congruency is proved