Is the quadrilateral 𝐴𝐵𝐶𝐷 cyclic?
We can begin by reminding ourselves that a cyclic quadrilateral is a quadrilateral with all four vertices inscribed on a circle. One property of cyclic quadrilaterals is that opposite angles are supplementary. We can check if a quadrilateral is cyclic by checking if opposite angles are supplementary. So let’s have a closer look at the figure that we’re given. The angle which is opposite to this given angle of 𝐴𝐵𝐶 would be the angle at 𝐷. If the angle at 𝐷 and this angle at 𝐵 add to 180 degrees, then 𝐴𝐵𝐶𝐷 would be cyclic.
So let’s see if we can indeed work out the measure of this angle. We are given that the measure of angle 𝐹𝐶𝐷 is 49 degrees. And we can observe that the angle measure at 𝐸𝐶𝐹 is marked as congruent. It’s also 49 degrees. The total angle measure then of angle 𝐸𝐶𝐷 will be 49 degrees plus 49 degrees, which is 98 degrees. We can then use the fact that we have a pair of parallel lines. And so angle 𝐴𝐷𝐶 is alternate to angle 𝐸𝐶𝐷. It’s also 98 degrees
Let’s remember that we’re checking if opposite angles are supplementary. Well, when we add together 98 degrees and 82 degrees, we do indeed get 180 degrees. So that means that opposite angles are supplementary. And so we can give the answer yes, since 𝐴𝐵𝐶𝐷 is cyclic.