Find the value of 𝑥.
Let’s have a look at the diagram we’ve been given. It consists of a triangle 𝐴𝐵𝐶 inside a circle. We’ve been given one angle in the triangle. It’s 50 degrees. And 𝑥 is the measure of the other angle. We’ve also been given the measure of the arc 𝐵𝐶. It’s 168 degrees. This arc 𝐵𝐶 is the intercepted arc of the inscribed angle 𝐵𝐴𝐶.
Let’s recall what we know about the measures of intercepted arcs and their inscribed angles. We recall that if an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. So the measure of the angle 𝐵𝐴𝐶 is equal to a half of the measure of the arc 𝐵𝐶.
We might not be able to work out angle 𝑥 straight away. But using this result, we can work out the measure of angle 𝐵𝐴𝐶. It’s equal to a half of 168 degrees or a half multiplied by 168 degrees, which is 84 degrees.
Let’s look at our diagram again. We’ve now filled in one more angle in this triangle, which means in triangle 𝐴𝐵𝐶, we know two of the angles. And we wish to calculate the third. We know that the angle sum in any triangle is 180 degrees. So if we know the other two angles, we can subtract them from 180 to find the value of 𝑥. 𝑥 is equal to 180 minus 84 minus 50, which is 46.
The value of 𝑥, found by recalling first of all that the measure of an inscribed angle in a circle is one-half the measure of its intercepted arc and then using the fact that the angle sum in a triangle is 180 degrees, is 46.