Question

question 9 of 25
true or false? if a parallelogram is inscribed in a circle, it must be a rectangle.
a. true
b. false

Explanation:

1. Recall the property of cyclic quadrilaterals: A quadrilateral inscribed in a circle (cyclic quadrilateral) has the sum of each pair of opposite angles equal to \(180^\circ\) (supplementary).
2. For a parallelogram, opposite angles are equal. Let the angles of the parallelogram be \(\angle A=\angle C\) and \(\angle B = \angle D\).
3. Since it is cyclic, \(\angle A+\angle C=180^\circ\) and \(\angle B+\angle D = 180^\circ\). But \(\angle A=\angle C\), so \(\angle A+\angle A=180^\circ\), which implies \(\angle A = 90^\circ\). Similarly, \(\angle B=90^\circ\).
4. A parallelogram with one right angle is a rectangle (all angles are right angles in a rectangle). So a parallelogram inscribed in a circle must be a rectangle.

Answer:

A. True