Given that the measure of angle 𝐵𝐶𝑀 equals 49 degrees, find the measure of angle 𝐵𝑀𝐶.
Let’s begin by filling in the angle measurements, being careful that we have got these values in the correct place. We are given that the measure of angle 𝐵𝐶𝑀 is 49 degrees. And the angle that we need to calculate is angle 𝐵𝑀𝐶. As we have a triangle, we could use the fact that the interior angle measures in a triangle add up to 180 degrees. However, we can’t use this property straightaway because we also don’t know the angle measure of angle 𝐶𝐵𝑀.
So let’s apply some reasoning about the fact that this triangle is created in a circle. The line segments 𝑀𝐶 and 𝑀𝐵 are both radii of the circle. And that’s helpful because that tells us that these two line segments will be congruent. This allows us to recognize that triangle 𝐵𝐶𝑀 is in fact an isosceles triangle. An isosceles triangle has two sides of equal length and two equal angle measures. So angle 𝐶𝐵𝑀 will be equal to angle 𝐵𝐶𝑀. In other words, they will both be 49 degrees.
Now, we can use the property that the interior angles in a triangle add up to 180 degrees. That means that the three angles of 49 degrees, 49 degrees, and the measure of angle 𝐵𝑀𝐶 must add to give 180 degrees. We can simplify 49 degrees plus 49 degrees to give us 98 degrees. And then we can subtract 98 degrees from both sides of the equation, which gives us the answer that the measure of angle 𝐵𝑀𝐶 is 82 degrees.