Question Video: Using the Alternate Segment Theorem to Find the Measure of an Angle Mathematics

Given that ๐‘šโˆ ๐ถ๐ด๐ต = 76ยฐ, find the value of ๐‘ฅ.

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Video Transcript

Given that the measure of angle ๐ถ๐ด๐ต is 76 degrees, find the value of ๐‘ฅ.

Letโ€™s begin by adding to our diagram what weโ€™ve been told, that the measure of the angle ๐ถ๐ด๐ต is 76 degrees. Now, angle ๐ถ๐ด๐ต is here. So how does that help us calculate the value of ๐‘ฅ? Well, itโ€™s important to realise that the lines joining ๐ถ and ๐ด and the lines joining ๐ต and ๐ด are tangents to the circle. We know that tangents to a circle that meet at a point are of equal length. And so we can see that triangle ๐ถ๐ด๐ต is actually an isosceles triangle.

We recall then that the base angles in an isosceles triangle are equal in size. And weโ€™ll use the fact that angles in a triangle add up to 180 degrees. Weโ€™re going to use these facts to calculate the size of angle ๐ด๐ถ๐ต. To do so, we subtract 76 from 180 degrees. And then, we divide this in two. And thatโ€™s because angle ๐ด๐ถ๐ต and ๐ด๐ต๐ถ are equal in size. 180 minus 76 divided by two is 52 degrees.

And now, we look really carefully at the shape weโ€™ve been given. We have a triangle inscribed in a circle. That means the radius at point ๐ถ. And so we can use the alternate segment theorem. Another way of saying this is that, in any circle, the angle between a chord and a tangent through one of the end points of the chord is equal to the angle in the alternate segment. And essentially, this means that the measure of the angle ๐ต๐ท๐ถ is 52 degrees.

So angle ๐‘ฅ is 52.

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