WEBVTT
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Determine the measure of angle 𝐵𝐶𝐷.
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So the first thing I’ve done is colored in the angle that we’re looking for.
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That’s angle 𝐵𝐶𝐷.
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We know it’s that one because the 𝐶 is in the middle, so that’s where the angle is going to be.
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So how do we work out what this is?
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Well, what we’re gonna use is the fact that what we’ve got here is a cyclic quadrilateral.
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But what is a cyclic quadrilateral?
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Well, a cyclic quadrilateral is a quadrilateral that has all four vertices that touch the circumference of a circle.
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So we can see here that our four vertices, and that can mean corners, are all touching the circumference of our circle.
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And we also know that it’s a quadrilateral because it’s a four-sided shape.
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Okay, great.
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We know that it’s a cyclic quadrilateral, but how is this gonna help us?
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And we know another property of a cyclic quadrilateral.
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And that is that, in a cyclic quadrilateral, the sum of a pair of opposite angles adds to 180 degrees.
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Well, that’s really useful because it shows us how we can answer the question.
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However, how do we know that a pair of opposite angles adds to 180 degrees.
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Well, I’ll quickly run through why it is.
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Well, the first thing I do is I mark on the center point of our circle.
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And then what I do is I draw radii to each of the four vertices of our quadrilateral.
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Well, because we’ve got four radii drawn, we know they’re all going to be the same length.
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So therefore what it tells us is that we have, in fact, four isosceles triangles.
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So therefore we’re gonna have pairs of equal angles, which I’ve shown here.
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So we got 𝑤 and 𝑤, we got 𝑥 and 𝑥, 𝑦 and 𝑦, and 𝑧 and 𝑧.
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And they must be equal because they’re the base angles of an isosceles triangle.
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Well, we know that the sum of the angles 𝐴𝐵𝐶𝐷 must be equal to 360 degrees because it’s a quadrilateral.
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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.
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So therefore two 𝑤 plus two 𝑥 plus two 𝑦 plus two 𝑧 must be equal to 360 degrees.
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So then if we divide through by two, we’re gonna get 𝑤 plus 𝑥 plus 𝑦 plus 𝑧 is equal to 180 degrees.
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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.
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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.
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So we’ve shown there how are pairs of opposite angles in a cyclic quadrilateral sum to 180 degrees.
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So now let’s use this and find out the angle we’re looking for.
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Well, the measure of angle 𝐵𝐶𝐷 is gonna be equal to 180 minus 78.
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And that’s because 𝐵𝐶𝐷 is opposite to 𝐵𝐴𝐷.
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So this is gonna give us an angle of 102 degrees.