Video: Finding the Size of an Angle inside a Triangle given the Size of the Two Other Angles Using the Circleโ€™s Tangent Properties

Given that ๐ด๐ต is a tangent to the circle ๐‘€ at ๐ด, and ๐‘šโˆ ๐‘€๐ต๐ธ = 151ยฐ, find ๐‘šโˆ ๐ด๐‘€๐ต.


Video Transcript

Given that ๐ด๐ต is a tangent to the circle ๐‘€ at ๐ด and the measure of angle ๐‘€๐ต๐ธ is 151 degrees, find the measure of angle ๐ด๐‘€๐ต.

Now in these questions, the first thing I always say is to mark on diagram what you know. Well, first of all, Iโ€™m gonna mark on 29 degrees for angle ๐ด๐ต๐‘€. And I found this because what I did that I took 151 degrees away from 180 degrees which gave me an answer of 29 degrees. And I could do this because the angles on a straight line equal 180 degrees.

So itโ€™s now that Iโ€™m actually gonna point out something thatโ€™s really key and important: whenever youโ€™re doing a question like this, make sure you give reasoning. This kind of questions that come at the exams always expect you to give reasons for your answers. So hereโ€™s my reasoning is that the angles on a straight line equal 180 degrees.

Okay, with this in mind, weโ€™ll move on and Iโ€™ll actually add on what else we know from our diagram. And next, we can mark on that angle ๐‘€๐ด๐ต is gonna be equal to 90 degrees. And thatโ€™s because itโ€™s a right angle. And thatโ€™s because thereโ€™s always a right angle between the radius of a circle and a tangent to a circle at point. To actually help you understand where this has come from, I actually gonna prove it using contradiction.

So Iโ€™ve drawn a little sketch here. So letโ€™s start by actually thinking- okay, ๐‘€๐ถ, letโ€™s suppose this isnโ€™t perpendicular to ๐ท๐ธ, which is our tangent. And from that, we then say โ€œokay, therefore ๐‘€๐ต is perpendicular.โ€ Well, therefore, we would say that angle ๐‘€๐ต๐ถ would be equal to 90 degrees and thus angle ๐‘€๐ถ๐ต must be acute because obviously we canโ€™t have two angles greater than 90 degrees in a triangle.

So because of this, we could say that as the greater angle is always opposite the longest side, then in this case, ๐‘€๐ต must be greater than ๐‘€๐ถ. But this doesnโ€™t sound right because we know that ๐‘€๐ถ is equal to ๐‘€๐ด because theyโ€™re both radius of a circle. So therefore, this would say that ๐‘€๐ต being greater than ๐‘€๐ถ is actually impossible. And therefore, this cannot be the case and therefore ๐‘€๐ต cannot be perpendicular.

So what weโ€™ve done here just gonna show you how the actual tangent to a circle against the radius is actually 90 degrees true contradiction. Okay, so now weโ€™ve done that. Letโ€™s get back on and solve the problem. So therefore, we can actually say that angle ๐ด๐‘€๐ต is gonna be equal to 180 degrees minus 90 degrees minus 29 degrees. And therefore, this is gonna be equal to 61 degrees.

But not forgetting, we need to give our reasoning. And our reasoning for this calculation is that the angles in a triangle are equal to 180 degrees. So therefore, weโ€™ve solved the problem and we found the measure of angle ๐ด๐‘€๐ต and it is 61 degrees.

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