True or False: All isosceles trapezoids are cyclic quadrilaterals.
Let’s begin by recalling that a trapezoid is a quadrilateral with one pair of parallel sides. An isosceles trapezoid is a special type of trapezoid that has the additional property that the two nonparallel sides or legs are equal in length. In the figure below, if we take the line segments 𝐵𝐶 and 𝐴𝐷 to be parallel, then that means that 𝐴𝐵𝐶𝐷 is an isosceles trapezoid. We then need to establish if isosceles trapezoids are cyclic quadrilaterals, that is, a quadrilateral which has all four vertices inscribed on a circle.
We can use the angle properties in a quadrilateral to help us determine if it’s cyclic or not. The diagonal 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 isosceles trapezoid. The diagonals of an isosceles trapezoid create two congruent triangles at the legs. The other two triangles at the bases are similar. Because we have these two congruent triangles, we know that the measure of angle 𝐴𝐵𝐷 will be equal to the measure of angle 𝐴𝐶𝐷.
The same is true for the angle measures of angle 𝐵𝐴𝐶 and angle 𝐵𝐷𝐶. Either of these pairs of angles would be sufficient to show that we have an angle created by the diagonal and side, which is equal in measure to the angle created by the other diagonal and opposite side. And so this isosceles trapezoid and all isosceles trapezoids are cyclic quadrilaterals. And so the statement in the question is true.