Video: Finding the Measure of an Angle in a Cyclic Quadrilateral Given the Measure of the Opposite Angle

Determine 𝑚∠𝐵𝐶𝐷.

03:25

Video Transcript

Determine the measure of angle 𝐵𝐶𝐷.

So the first thing I’ve done is colored in the angle that we’re looking for. That’s angle 𝐵𝐶𝐷. We know it’s that one because the 𝐶 is in the middle, so that’s where the angle is going to be. So how do we work out what this is? Well, what we’re gonna use is the fact that what we’ve got here is a cyclic quadrilateral. But what is a cyclic quadrilateral? Well, a cyclic quadrilateral is a quadrilateral that has all four vertices that touch the circumference of a circle. So we can see here that our four vertices, and that can mean corners, are all touching the circumference of our circle. And we also know that it’s a quadrilateral because it’s a four-sided shape.

Okay, great. We know that it’s a cyclic quadrilateral, but how is this gonna help us? And we know another property of a cyclic quadrilateral. And that is that, in a cyclic quadrilateral, the sum of a pair of opposite angles adds to 180 degrees. Well, that’s really useful because it shows us how we can answer the question. However, how do we know that a pair of opposite angles adds to 180 degrees. Well, I’ll quickly run through why it is.

Well, the first thing I do is I mark on the center point of our circle. And then what I do is I draw radii to each of the four vertices of our quadrilateral. Well, because we’ve got four radii drawn, we know they’re all going to be the same length. So therefore what it tells us is that we have, in fact, four isosceles triangles. So therefore we’re gonna have pairs of equal angles, which I’ve shown here. So we got 𝑤 and 𝑤, we got 𝑥 and 𝑥, 𝑦 and 𝑦, and 𝑧 and 𝑧. And they must be equal because they’re the base angles of an isosceles triangle.

Well, we know that the sum of the angles 𝐴𝐵𝐶𝐷 must be equal to 360 degrees because it’s a quadrilateral. So therefore we can say that 𝑤 add 𝑥 plus 𝑥 add 𝑦 plus 𝑦 add 𝑧 plus 𝑧 add 𝑤 must be equal to 360 degrees. So therefore two 𝑤 plus two 𝑥 plus two 𝑦 plus two 𝑧 must be equal to 360 degrees. So then if we divide through by two, we’re gonna get 𝑤 plus 𝑥 plus 𝑦 plus 𝑧 is equal to 180 degrees. So therefore angles 𝐴 plus 𝐶 are gonna be equal to 𝑤 plus 𝑥 plus 𝑦 plus 𝑧, which is gonna be equal to 180 degrees because it contains each of our components 𝑤, 𝑥, 𝑦, and 𝑧. And then the other opposite pair, 𝐵 and 𝐷, is gonna be equal to 𝑥 plus 𝑦 plus 𝑧 plus 𝑤, which again is gonna be equal to 180 degrees.

So we’ve shown there how are pairs of opposite angles in a cyclic quadrilateral sum to 180 degrees. So now let’s use this and find out the angle we’re looking for. Well, the measure of angle 𝐵𝐶𝐷 is gonna be equal to 180 minus 78. And that’s because 𝐵𝐶𝐷 is opposite to 𝐵𝐴𝐷. So this is gonna give us an angle of 102 degrees.

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