True or False: All trapezoids are cyclic quadrilaterals.
We can begin by recalling that a trapezoid is a quadrilateral with one pair of parallel sides. In the figure that we are given, if the line segments 𝐷𝐶 and 𝐴𝐵 are parallel, then we would have a trapezoid. What we need to do here is to establish if a trapezoid is cyclic or not. A cyclic quadrilateral is one which has all four vertices inscribed on a circle. In order to help us determine if it’s cyclic, let’s use the angle properties at the diagonals here to help us. This property tells us that if an angle created by a diagonal and side is equal in measure to the angle created by the other diagonal and opposite side, then the quadrilateral is cyclic.
Let’s consider this angle 𝐷𝐴𝐶, which is an angle created by a side and a diagonal. The angle created by the other diagonal and opposite side would be this angle 𝐷𝐵𝐶. But we can see even by eye that these two angle measures would not be the same. That would mean that this trapezoid is not a cyclic quadrilateral. In fact, there is only one type of trapezoid that is a cyclic quadrilateral. And that’s an isosceles trapezoid, which is a special type of trapezoid with the additional property that the two nonparallel sides are congruent.
So while it’s useful to note that isosceles trapezoids are cyclic quadrilaterals, we cannot say that all trapezoids are cyclic quadrilaterals. And so the answer to the statement is false.