What is the measure of angle 𝑀𝐴𝐵?
In this figure, we can notice that the angle 𝑀𝐴𝐵 appears here in the triangle. It might not immediately be obvious how we’re going to calculate this angle measure because we do have quite a few unknown angle measures. Let’s begin by seeing if we can calculate the acute angle of 𝐴𝑀𝐵. To do this, we’ll use the fact that the angle measures about a point sum to 360 degrees. So that means that this acute angle measure of 𝐴𝑀𝐵 must be equal to 360 degrees subtract 306 degrees, which leaves us with 54 degrees.
Next, we can use the fact that we have this triangle 𝐴𝑀𝐵 and the fact that the interior angles in a triangle add up to 180 degrees. However, we do have a bit of a problem because we don’t actually know the angle measure of angle 𝐴𝐵𝑀. We can observe though that the line segments 𝐴𝑀 and 𝐵𝑀 are both radii of the circle. And that means that they will be the same length. We can therefore say that triangle 𝐴𝐵𝑀 must be an isosceles triangle, which has two equal sides and two equal angle measures.
The two angles which will have equal angle measures are angle 𝑀𝐴𝐵 and angle 𝑀𝐵𝐴. Let’s define these both as 𝑥 degrees. We know that these three angle measures will add to give 180 degrees. We can therefore say that 54 degrees plus two 𝑥 degrees is equal to 180 degrees. Subtracting 54 degrees from both sides, we have two 𝑥 degrees is equal to 126 degrees. Dividing both sides of the equation, we have that 𝑥 degrees is equal to 63 degrees. Since we defined the measure of angle 𝑀𝐴𝐵 as 𝑥 degrees, then we can give the answer that this angle measure is 63 degrees.