Given that the measure of angle 𝐴𝐵𝐶 equals six 𝑥 plus 15 degrees and the measure of angle 𝐶𝐴𝐵 is 11𝑥 minus 10 degrees, find the value of 𝑥.
Let’s begin by adding our angles to the diagram. We’re told that the measure of angle 𝐴𝐵𝐶 is six 𝑥 plus 15 degrees. That’s the angle at the bottom of our diagram. Similarly, the measure of angle 𝐶𝐴𝐵 is 11𝑥 minus 10 degrees. That’s the angle at the top of our diagram. And it might seem at the moment like we don’t have enough information to solve the problem. But if we inspect the circle carefully, we notice that line 𝐴𝐵 passes through point 𝑀. So 𝐴𝐵 must be, in fact, the diameter of our circle. So we can use a special case of the inscribed angle theorem to find the third angle in our triangle.
We know that the angle subtended by the diameter is 90 degrees. Well, the angle in our triangle subtended by the diameter is 𝐴𝐶𝐵, so 𝐴𝐶𝐵 must be equal to 90 degrees. Secondly, we know that the interior angle sum of a triangle is 180 degrees, so we find the angle sum in our triangle. It’s 11𝑥 minus 10 plus six 𝑥 plus 15 plus 90, and that equals 180 degrees. Let’s simplify by collecting like terms. When we do, 11𝑥 plus six 𝑥 is 17𝑥. Then we find the sum of negative 10, 15, and 90, giving us 95. So in degrees, the interior angle sum in our triangle is 17𝑥 plus 95.
This, of course, is equal to 180. Now we’re leaving the degrees symbol out here because it can be a little bit confusing adding it in all the different locations. We’re going to solve for 𝑥 by subtracting 95 from both sides. 180 minus 95 is 85. So 17𝑥 equals 85. Finally, we divide through by 17. So 𝑥 is 85 divided by 17, which is equal to five. And so we found the value of 𝑥. It’s five.