Question Video: Using the Properties of Cyclic Quadrilaterals to Verify Whether the Given Trapezoid Is Cyclic Mathematics

Video Transcript

Is the trapezoid 𝐴𝐵𝐶𝐷 a cyclic quadrilateral?

Let’s start by reminding ourselves that a cyclic quadrilateral is a quadrilateral whose vertices are inscribed on a circle. There are a few different angle properties that we can use to prove if a quadrilateral is cyclic or not.

Given that we have the diagonals marked, we may choose to use the property 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. For example, if we could show that the angle 𝐴𝐷𝐵 was equal in measure to angle 𝐴𝐶𝐵, or indeed if we had another angle pair, for example, angle 𝐷𝐴𝐶 equal to angle 𝐷𝐵𝐶, then the quadrilateral would be cyclic.

So let’s see if we can work out the angle measure of 𝐵𝐶𝐴. Because we’re given that 𝐴𝐵𝐶𝐷 is a trapezoid, that means there’ll be one set of parallel sides. And we can see them here. Sides 𝐵𝐶 and 𝐴𝐷 are parallel. Therefore, angle 𝐷𝐵𝐶 is alternate to this given angle of 𝐴𝐷𝐵. And so they’re both equal to 84 degrees.

Using the fact that the interior angles in a triangle add up to 180 degrees will allow us to find our unknown angle. We can write that the three angle measures of 84 degrees plus 52 degrees plus the measure of angle 𝐵𝐶𝐴 is equal to 180 degrees. We can simplify this and then subtract 136 degrees from both sides, giving us that the measure of angle 𝐵𝐶𝐴 is 44 degrees.

We can now really easily see that this angle measure of 𝐵𝐶𝐴 is not equal to the angle measure at 𝐵𝐷𝐴. Therefore, we can give the answer no, trapezoid 𝐴𝐵𝐶𝐷 is not a cyclic quadrilateral.