Question Video: Using Properties of Cyclic Quadrilaterals to Solve a Problem Mathematics

Video Transcript

Given that the measure of angle 𝐷𝐶𝐻 is 92 degrees, the measure of angle 𝐵𝐴𝐷 is four 𝑥 degrees, and the measure of angle 𝐶𝐷𝐴 is two 𝑦 degrees, find 𝑥 plus 𝑦.

So let’s have a look at this diagram we’ve been given here. We have a quadrilateral within a circle. Each of the four vertices of the quadrilateral touch the circumference of this circle. So this is a cyclic quadrilateral. We’ve been told that the measure of angle 𝐷𝐶𝐻 is 92 degrees. We’ve also been told that the measure of angle 𝐵𝐴𝐷 is four 𝑥 degrees and the measure of angle 𝐶𝐷𝐴 is two 𝑦 degrees.

So when answering a question like this, it’s always good to mark on the diagram any angles that you’ve already been given in the question. We’ve been asked to find the value of 𝑥 plus 𝑦. So we’re going to need to find the value of 𝑥, and we’re going to need to find the value of 𝑦. So let’s use the fact that we’ve got a cyclic quadrilateral here to calculate the values for these angles.

To start, 𝐵𝐶𝐻 is a straight line and 𝐷𝐶𝐻 is 92 degrees. And we know that angles on a straight line add up to 180 degrees. So we can calculate 𝐵𝐶𝐷 to be 180 minus 92. That gives us 88. So angle 𝐵𝐶𝐷 is 88 degrees. Now notice that we have either a value or an expression for all of the angles within the cyclic quadrilateral. These angles are 82 degrees, 88 degrees, two 𝑦 degrees, and four 𝑥 degrees.

Of course, we know that angles inside a quadrilateral add up to 360 degrees. But we can’t use that fact to find both the unknown values 𝑥 and 𝑦. So what else do we know that might be able to help us? Well, one property of cyclic quadrilaterals is that opposite angles add up to 180 degrees. So what we’re saying is that four 𝑥 plus 88 must equal 180. And also 82 plus two 𝑦 must also equal 180.

So starting with four 𝑥 plus 88 equals 180 and subtracting 88 from both sides, we have that four 𝑥 equals 92. And then dividing both sides by four gives us that 𝑥 equals 23. Now if we consider 82 plus two 𝑦 equals 180, subtracting 82 from both sides of the equation gives us that two 𝑦 equals 98. And therefore, 𝑦 equals 49.

But we’re still not done with the question because we were asked to find 𝑥 plus 𝑦. So we know that 𝑥 plus 𝑦 equals 23 plus 49. And that gives us 72.