Why a Quadrilateral with Exactly One Right Angle is a Parallelogram: Proof and Properties

A quadrilateral with exactly one right angle is a parallelogram

In order to prove that a quadrilateral with exactly one right angle is a parallelogram, we need to demonstrate that it satisfies the definition and properties of a parallelogram

In order to prove that a quadrilateral with exactly one right angle is a parallelogram, we need to demonstrate that it satisfies the definition and properties of a parallelogram.

A parallelogram is a quadrilateral with two pairs of parallel sides. Additionally, it has opposite angles that are congruent (equal) and opposite sides that are also congruent.

Let’s start by considering a quadrilateral ABCD with exactly one right angle. Without loss of generality, let’s assume that angle A is the right angle.

To prove that ABCD is a parallelogram, we need to show the following properties:

1. Opposite sides are parallel:
Since ABCD has a right angle, we can conclude that angles A and C are supplementary (their measures add up to 180 degrees). It is known that if a pair of angles is supplementary, then the sides opposite these angles are parallel. Hence, side AB is parallel to side CD, and side AD is parallel to side BC.

2. Opposite sides are congruent:
Since ABCD is a quadrilateral with a right angle and both pairs of opposite sides are parallel, we can use various methods to show that opposite sides are congruent. For example, we can use the Pythagorean theorem if we have enough information about the lengths of the sides.

3. Opposite angles are congruent:
Since ABCD has a right angle, we know that angle A is congruent to angle C (both are 90 degrees). Moreover, we can use the property that angles that are supplementary to the same angle are congruent. Therefore, angles A and B are supplementary, as well as angles C and D. This means that opposite angles in ABCD are congruent.

Thus, we have shown that the quadrilateral ABCD with exactly one right angle satisfies all properties of a parallelogram. Therefore, we can conclude that a quadrilateral with exactly one right angle is indeed a parallelogram.

More Answers: