Chapter 12.2, Problem 16E

To determine

To prove:

TheGiven on graph $\Delta ABG\cong \Delta DCF$ .

Explanation of Solution

Given:

  $\angle A\cong \angle D\text{and}\angle B\cong \angle C$

Concept used:

Two geometric figures are congruent if a rigid motion or a composition of rigid motions maps one of the figures onto other.

Calculation:

The pairs of congruent corresponding angles

Vertivalangles are congruent

  $\angle CFD\cong \angle BGA$

  $\angle A\cong \angle D\text{and}\angle B\cong \angle C$

A midpoint cuts a segment into two congruent

  $\begin{array}{l}\overline{AB}\cong \overline{DC}\\ \overline{EB}\cong \overline{EC}\\ \overline{EF}\cong \overline{EG}\\ \overline{AF}\cong \overline{DG}\end{array}$

Segment addition theorem

  $\begin{array}{l}CF=EC+EF\\ BG=EB+EG\end{array}$

Since

  $\begin{array}{l}EB=EG=EC=EF\\ CF=EC+EF=BG\end{array}$

So, $\overline{CF}\cong \overline{BG}$

  $\begin{array}{l}AG=AF+FG\\ DF=FG+DG\end{array}$

Since

  $\begin{array}{l}AF=DG\\ AG=AF+DG=DF\end{array}$

So, $\overline{AG}\cong \overline{DF}$

  $\Delta ABG\cong \Delta DCF$

Therefore,

  $\Delta ABG\cong \Delta DCF$