# Video: Finding the Size of an Angle inside a Triangle given the Size of the Two Other Angles Using the Circle’s Tangent Properties

Given that 𝐴𝐵 is a tangent to the circle 𝑀 at 𝐴, and 𝑚∠𝑀𝐵𝐸 = 151°, find 𝑚∠𝐴𝑀𝐵.

03:09

### Video Transcript

Given that 𝐴𝐵 is a tangent to the circle 𝑀 at 𝐴 and the measure of angle 𝑀𝐵𝐸 is 151 degrees, find the measure of angle 𝐴𝑀𝐵.

Now in these questions, the first thing I always say is to mark on diagram what you know. Well, first of all, I’m gonna mark on 29 degrees for angle 𝐴𝐵𝑀. 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 it’s now that I’m actually gonna point out something that’s really key and important: whenever you’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 here’s my reasoning is that the angles on a straight line equal 180 degrees.

Okay, with this in mind, we’ll move on and I’ll actually add on what else we know from our diagram. And next, we can mark on that angle 𝑀𝐴𝐵 is gonna be equal to 90 degrees. And that’s because it’s a right angle. And that’s because there’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 I’ve drawn a little sketch here. So let’s start by actually thinking- okay, 𝑀𝐶, let’s suppose this isn’t perpendicular to 𝐷𝐸, which is our tangent. And from that, we then say “okay, therefore 𝑀𝐵 is perpendicular.” Well, therefore, we would say that angle 𝑀𝐵𝐶 would be equal to 90 degrees and thus angle 𝑀𝐶𝐵 must be acute because obviously we can’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, 𝑀𝐵 must be greater than 𝑀𝐶. But this doesn’t sound right because we know that 𝑀𝐶 is equal to 𝑀𝐴 because they’re both radius of a circle. So therefore, this would say that 𝑀𝐵 being greater than 𝑀𝐶 is actually impossible. And therefore, this cannot be the case and therefore 𝑀𝐵 cannot be perpendicular.

So what we’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 we’ve done that. Let’s get back on and solve the problem. So therefore, we can actually say that angle 𝐴𝑀𝐵 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, we’ve solved the problem and we found the measure of angle 𝐴𝑀𝐵 and it is 61 degrees.