Full solution

1

. Given: Parallelogram ABFE Parallelogram EFCD

\overline{A D} \perp \overline{D C}

\newline

Prove:

A B C D

is a rectangle

Given information: We are given that $ABFE$ and $EFCD$ are parallelograms. By the definition of a parallelogram, opposite sides are equal and parallel.
Properties of ABFE: Since ABFE is a parallelogram, we have $AB \parallel FE$ and $AB = FE$ .
Properties of EFCD: Similarly, since EFCD is a parallelogram, we have $EF \parallel CD$ and $EF = CD$ .
Consecutive angles in parallelograms: From the properties of parallelograms, we also know that consecutive angles in a parallelogram are supplementary. Therefore, angle $AFE$ + angle $EFD$ = $180$ degrees.
Perpendicularity of $AD$ and $DC$ : We are given that line segment $AD$ is perpendicular to line segment $DC$ , which means angle $ADC$ is a right angle ( $90$ degrees).
Supplementary angles: Since angle $AFE$ and angle $EFD$ are supplementary and angle $ADC$ is a right angle, angle $AFE$ must also be a right angle because $\angle AFE + 90^\circ = 180^\circ$ .
Perpendicularity of $AB$ and $FE$ : Now we have two consecutive right angles in the shape $ABFE$ , which means that $AB$ is perpendicular to $FE$ .
Perpendicularity of $CD$ and $FE$ : Since $AB$ is perpendicular to $FE$ and $AB \parallel CD$ (because $EFCD$ is a parallelogram), it follows that $CD$ is also perpendicular to $FE$ .
Conclusion of properties: We now have that $AB$ is perpendicular to $FE$ and $CD$ is perpendicular to $FE$ , which means $AB$ is parallel and equal to $CD$ , and both are perpendicular to $AD$ and $DC$ .
Definition of a rectangle: Therefore, we have shown that all four angles in quadrilateral $ABCD$ are right angles, and opposite sides are equal and parallel.
Definition of a rectangle: Therefore, we have shown that all four angles in quadrilateral $ABCD$ are right angles, and opposite sides are equal and parallel.By the definition of a rectangle, a quadrilateral with four right angles and opposite sides that are equal and parallel is a rectangle.