In the figure below, the measure of angle 𝐴𝐷𝐸 equals 26.9 degrees, the measure of angle 𝐴𝐸𝐵 equals 82.9 degrees, and the measure of angle 𝐴𝐵𝐸 equals 59.1 degrees. Is 𝐴𝐵𝐶𝐷 a cyclic quadrilateral?
We can recall that a cyclic quadrilateral is a quadrilateral that has all four vertices inscribed on a circle. One angle property we can use to prove a quadrilateral is cyclic is 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.
So let’s have a look at this quadrilateral. And we’ll begin by filling in the angle measures we were given. Here are the angle measures then. 𝐴𝐷𝐸 is 26.9 degrees, 𝐴𝐸𝐵 is 82.9 degrees, and 𝐴𝐵𝐸 is 59.1 degrees. We might notice then that within this triangle of 𝐴𝐵𝐸, we have two angle measures and one unknown. Using the fact that the interior angles in a triangle add up to 180 degrees will allow us to calculate the measure of angle 𝐸𝐴𝐵. So we’ll have 82.9 degrees plus 59.1 degrees plus the measure of angle 𝐸𝐴𝐵 is equal to 180 degrees.
Simplifying 82.9 degrees plus 59.1 degrees gives us 142 degrees. Subtracting 142 degrees from both sides leaves us with the measure of angle 𝐸𝐴𝐵 equal to 38 degrees. At this point, however, we still don’t have the measures of a pair of angles made at the diagonals. For example, if we consider this angle at 𝐸𝐴𝐵 and see if it’s equal to the angle made by the other diagonal and opposite side, that is, angle 𝐶𝐷𝐵, then we could check if the quadrilateral is cyclic. An alternative pair of angles we could check would be angles 𝐷𝐶𝐴 and angle 𝐷𝐵𝐴.
For either angle pair, we’ll need to keep going and see what other angles we can work out in this figure. The fact that 𝐷𝐵 is a straight line will allow us to calculate this angle measure of 𝐴𝐸𝐷. So the measure of angle 𝐴𝐸𝐷 is equal to 180 degrees subtract 82.9 degrees, which leaves us with 97.1 degrees. Observing then that we have this triangle 𝐴𝐸𝐷 and two known angles will allow us to calculate the third angle 𝐷𝐴𝐸. Because these three angles will add to 180 degrees, we can write that 26.9 degrees plus 97.1 degrees plus the measure of angle 𝐷𝐴𝐸 is equal to 180 degrees.
We can simplify this and then subtract 124 degrees from both sides of the equation. And so the measure of angle 𝐷𝐴𝐸 is 56 degrees.
In order to do any further calculations in this figure, we’ll need to observe some important markings. We are given that the line segments 𝐴𝐷 and 𝐶𝐷 are congruent. When we consider these line segments as part of the triangle 𝐴𝐶𝐷, then that means that triangle 𝐴𝐶𝐷 is an isosceles triangle. An isosceles triangle has two equal sides and two equal angle measures. And that means that the measure of angle 𝐷𝐴𝐶 must be equal to the measure of angle 𝐷𝐶𝐴. And so they’ll both be 56 degrees.
So remember that we’ve done all these angle measurement calculations in order to see if we have a pair of equal angles at the diagonals. And we do now have a pair of angles which we can compare: angle 𝐴𝐵𝐷 and angle 𝐴𝐶𝐷. And we can see that of course 56 degrees is not equal to 59.1 degrees. This means that an angle created by a diagonal and side is not equal to the angle created by the other diagonal and opposite side. And so we can give the answer no, because we have proved that 𝐴𝐵𝐶𝐷 is not a cyclic quadrilateral.