Is 𝐴𝐵𝐶𝐷 a cyclic quadrilateral?
Let’s recall that a cyclic quadrilateral is a four-sided polygon whose vertices are inscribed on a circle. We can prove that a quadrilateral is cyclic or not by using a few different properties. However, we can observe in 𝐴𝐵𝐶𝐷 that we’re given the diagonals. 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. If these angle measures are not equal, then it is not a cyclic quadrilateral.
So, let’s look at this angle 𝐶𝐷𝐵. It’s an angle created by a diagonal and a side. The angle created by the other diagonal and opposite side would be this angle 𝐶𝐴𝐵. And of course 82 degrees is not equal to 78 degrees. Therefore, the measure of angle 𝐶𝐷𝐵 is not equal to the measure of angle 𝐶𝐴𝐵.
We can then give the answer no, since 𝐴𝐵𝐶𝐷 is not a cyclic quadrilateral.