Chapter 5, Problem 31CUR

To determine

To prove: $\overline{EF}\cong \overline{FG}$

Explanation of Solution

Given information: $\overline{AD}\cong \overline{BC},\overline{AD}\parallel \overline{BC}$

Proof:

  

In the given figure,

  $\overline{AD}\cong \overline{BC},\overline{AD}\parallel \overline{BC}$

So, ABCD is a parallelogram.

(A quadrilateral in which a pair of sides is equal and parallel is a parallelogram)

Consider $\Delta AFE,\Delta CFG$

  $\angle GCF\cong \angle EAF$ (If lines are parallel then each pair of alternate interior angles are congruent)

  $\overline{CF}\cong \overline{AF}$ (Diagonals of a parallelogram bisect each other)

  $\angle GFC\cong \angle EFA$ (Vertically opposite angles)

So,

  $\Delta CFG\cong \Delta AFE$ (According to ASA congruence criteria, if two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle then the two triangles are congruent)

Therefore,

  $\overline{FG}\cong \overline{EF}$ (Corresponding parts of congruent triangles are congruent)