# Question Video: Finding the Measure of an Angle given Its Arc’s Measure Using Another Inscribed Angle Mathematics

Given that 𝑚∠𝐴𝐵𝐶 = 30.5°, find 𝑚∠𝐵𝐷𝐶.

02:30

### Video Transcript

Given that the measure of angle 𝐴𝐵𝐶 is equal to 30.5 degrees, find the measure of angle 𝐵𝐷𝐶.

So the first thing we can do to start to solve this problem is how we look at our angle 𝐴𝐵𝐶. Well, we know that the measure of an inscribed angle is half the measure of that that subtends it. So what does this mean in practice? It means that the angle 𝐴𝐵𝐶 is half of the angle that is at the arc 𝐶𝐴.

So therefore, we can say that the measure of angle 𝐴𝐵𝐶 is equal to half the measure of the arc 𝐶𝐴. And as we know that the measure of angle 𝐴𝐵𝐶 is equal to 30.5, we can say that 30.5 is equal to half the measure of the arc 𝐶𝐴. So therefore, if we multiply each side of the equation by two, we can get that the measure of the arc 𝐶𝐴 is gonna be equal to 61 degrees.

But how does this help us? Well, first of all, we’re gonna look at another relationship. We can see that 𝐴𝐵, through the line 𝐴𝐵, passes through 𝑀, which is the centre of our circle. So therefore, the arc 𝐴 to 𝐵 must be equal to 180 degrees. So therefore, we can say that the measure of the arc from 𝐶𝐴𝐵 is gonna be equal to the measure of the arc 𝐶𝐴 plus 180 because it’d be plus the arc 𝐴 to 𝐵. So therefore, we can say that the measure of the arc from 𝐶𝐴𝐵 is gonna be equal to 241 degrees cause it’s 61 plus 180.

Well now, if we take a look back at the question, what we want to do is find the measure of the angle 𝐵𝐷𝐶. Well, if we look at 𝐵𝐷𝐶, then the angle 𝐵𝐷𝐶 is actually subtended by the arc 𝐶𝐴𝐵. So therefore, we can say that the measure of the angle 𝐵𝐷𝐶 is gonna be equal to half of the arc 𝐶𝐴𝐵. So therefore, we can say that the measure of angle 𝐵𝐷𝐶 is gonna be equal to a half multiplied by 241.

So therefore, we can say that the measure of angle 𝐵𝐷𝐶 is gonna be equal to 120.5 degrees. It’s cause that was half of 241.

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