Chapter 12.7, Problem 12E

To determine

To prove: The statement $\angle 1\cong \angle 2$ .

Answer to Problem 12E

The proof that $\angle 1\cong \angle 2$ is stated below.

Explanation of Solution

Consider $\Delta ABE$ and $\Delta CBE$ .

  $\begin{array}{c}\angle ABE=\angle CBE\\ \overline{BE}=\overline{BE}\\ \angle AEB=\angle CEB\end{array}$

Thus, by ASA congruence theorem.

  $\Delta ABE\cong \Delta CBE$

Consider $\Delta AFE$ and $\Delta CDE$ .

  $\begin{array}{c}\overline{AE}=\overline{CE}\\ \angle FAE=\angle DCE\\ \overline{FA}=\overline{DC}\end{array}$

Thus, by SAS congruence theorem.

  $\Delta AFE\cong \Delta CDE$

So, by CPCT,

  $\angle 1\cong \angle 2$

Hence, proved.