Chapter 5, Problem 23RP

To determine

Prove that the $ABCD$ is a parallelogram.

Answer to Problem 23RP

Proved, $ABCD$ is a parallelogram.

Explanation of Solution

Given information:

  

  $FEGH$ is a parallelogram, $\overline{AE}\cong \overline{BF}\cong \overline{CG}\cong \overline{DH}$ .

Calculation:

Construct the figure as per the given statement,

  

We know that the opposite angle of the parallelogram are congruent,

  $\angle EHG\cong \angle EFGand\angle FEH\cong \angle FGH$

Angles supplementary to congruent angles are congruent,

  $\angle CHD\cong \angle AFBand\angle AED\cong \angle BGC$

Opposite sides of the parallelogram are congruent,

  $\overline{HE}\cong \overline{FG}and\overline{EF}\cong \overline{HG}$

We have given, $\overline{AE}\cong \overline{BF}\cong \overline{CG}\cong \overline{DH}$

Addition congruent segments are,

  $\begin{array}{l}AE+EF=CG+HG\Rightarrow \overline{AF}\cong \overline{CH}\\ BF+FG=DH+HE\Rightarrow \overline{BG}\cong \overline{DE}\end{array}$

  $\begin{array}{l}\Delta ABF\cong \Delta CDH,\Delta ADE\cong \Delta CBG\left[SASCongruence\right]\\ \overline{AB}\cong \overline{CD},\overline{AD}\cong \overline{CB}\end{array}$

A quadrilateral with two pairs of opposite congruent sides is a parallelogram,

Hence, proved $ABCD$ is a parallelogram.