Is there a circle passing through the vertices of the quadrilateral 𝐴𝐵𝐶𝐷?
If, in fact, there is a circle which passes through the vertices of 𝐴𝐵𝐶𝐷, then we would say that this is a cyclic quadrilateral. We can use the angle properties to check if the quadrilateral is cyclic. We can recall that if the angle made with a diagonal and side is equal in measure to the angle made with the other diagonal and opposite side, then the quadrilateral is cyclic. We can draw in this fourth line segment of the quadrilateral 𝐴𝐵𝐶𝐷, the side 𝐴𝐷. An angle that is made by a diagonal and side is the angle 𝐶𝐴𝐵. The angle which is created by the other diagonal and the opposite side would be the angle 𝐶𝐷𝐵.
However, we aren’t given the measure of this angle, but let’s see if we can work it out. Using triangle 𝐶𝐵𝐷, we can recall that the interior angle measures in a triangle sum to 180 degrees. That means that the three angle measures of 54 degrees, 79 degrees, and the unknown angle measure of 𝐶𝐷𝐵 must add to give 180 degrees. We can then add 54 degrees and 79 degrees to give us 133 degrees and then subtract 133 degrees from both sides. This gives us that the measure of angle 𝐶𝐷𝐵 is 47 degrees. And so now we can say that the angle made with the diagonal and side is equal in measure to the angle made with the other diagonal and opposite side. And so that means that 𝐴𝐵𝐶𝐷 is a cyclic quadrilateral.
We can therefore give the answer yes, there is a circle passing through the vertices of the quadrilateral 𝐴𝐵𝐶𝐷. We could even draw this circle if we wished.