Is 𝐴𝐵𝐶𝐷 a cyclic quadrilateral?
We can remember that a cyclic quadrilateral is a quadrilateral which has all four vertices inscribed on a circle. We can prove if a quadrilateral is cyclic or not by checking some angle properties. Given that we have the diagonals marked on this quadrilateral, then that might give us a clue. We can check 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.
In the figure, we are given the measure of this angle 𝐴𝐷𝐵. This is an angle, which is made from a diagonal and side. The angle which is created by the other diagonal and the opposite side would be here, angle 𝐴𝐶𝐵. If we could demonstrate that the measure of this angle was the same as the measurement of angle 𝐴𝐷𝐵, then we would have a cyclic quadrilateral. Let’s see if we can calculate this angle measure.
Let’s use the fact that it’s part of this triangle 𝐸𝐶𝐵 to help. We should notice that we have a right angle here at angle 𝐴𝐸𝐵. And because the angles on a straight line sum to 180 degrees, then we know that the angle measure of 𝐵𝐸𝐶 is also 90 degrees. So now within this triangle 𝐵𝐸𝐶, we can use the fact that the interior angles of triangle add up to 180 degrees. And so we’ll have 63 degrees plus 90 degrees plus the measure of angle 𝐵𝐶𝐸 must be equal to 180 degrees. Adding 63 degrees and 90 degrees gives us 153 degrees, and subtracting 153 degrees from both sides gives us that the measure of angle 𝐵𝐶𝐴 is 27 degrees.
So now if we compare the angles made at the diagonals, we have this angle measure of 27 degrees and this angle of 38 degrees. Of course, 27 degrees is not equal to 38 degrees. Therefore, an angle created by a diagonal and side is not equal to an angle created by the other diagonal and opposite side. And so 𝐴𝐵𝐶𝐷 is not a cyclic quadrilateral. And so we can give the answer no.
There is an alternative angle pair that we could also have checked. The angle 𝐶𝐵𝐷 is an angle made by a diagonal and side. The angle created by the other diagonal and opposite side would be this angle at 𝐶𝐴𝐷. We could have established that this angle at 𝐴𝐸𝐷 is a right angle and then use the fact that the angles in a triangle add up to 180 degrees to work out the unknown angle. We would have then been able to calculate that the measure of angle 𝐶𝐴𝐷 is 52 degrees.
This time, we would be able to show that 63 degrees is not equal to 52 degrees. And so this would show that 𝐴𝐵𝐶𝐷 is not a cyclic quadrilateral. But we don’t need to show that there are two angle pairs which are not equal. It’s sufficient just to have one pair of angles at the diagonals which are not equal in order to prove that the quadrilateral is not cyclic.