Question

2.12.2 from conjecture to proof
here is another conjecture: if a parallelogram has (at least)
one right angle, then it is a rectangle.
sketch a picture to convince yourself it is true. label all of the
points and show any markings that are given.
using the picture, give a brief idea of how you can prove the
conjecture you chose below.
a helpful fact: adjacent angles (angles next to each other) in a
parallelogram are supplementary (add up to 180°)

Explanation:

Let's assume parallelogram \(ABCD\) with \(\angle A = 90^{\circ}\). In a parallelogram, \(AD\parallel BC\). Since adjacent angles in a parallelogram are supplementary, \(\angle A+\angle B = 180^{\circ}\). Substituting \(\angle A = 90^{\circ}\), we get \(90^{\circ}+\angle B=180^{\circ}\), so \(\angle B = 90^{\circ}\). Also, opposite angles in a parallelogram are equal. So \(\angle C=\angle A = 90^{\circ}\) and \(\angle D=\angle B = 90^{\circ}\). A quadrilateral with four right - angles is a rectangle.

Answer:

If a parallelogram has one right angle, using the properties of parallelograms (adjacent angles are supplementary and opposite angles are equal), we can show all four angles are right angles, thus proving it is a rectangle.