Question Video: Finding the Measure of an Angle in a Triangle given the Other Two Angles’ Measures Using the Tangents’ Properties Mathematics

Video Transcript

Given that the measure of angle 𝑀𝐴𝐶 equals 36 degrees, determine the measure of angle 𝐵𝐴𝑀 and the measure of angle 𝐴𝑀𝐶.

We want to know the measure of angle 𝐵𝐴𝑀 and the measure of angle 𝐴𝑀𝐶. Let’s start by identifying what we know. We know that the measure of angle 𝑀𝐴𝐶 is 36 degrees, so we can add that to the diagram. We’ll also wanna list other information we can identify from the diagram. We know that line segment 𝐴𝐶 and the line segment 𝐴𝐵 are tangent to the circle 𝑀 since each segment intersects the circle at exactly one point. Line segment 𝐴𝐶 and line segment 𝐴𝐵 intersect at point 𝐴. From there, we say that line segment 𝑀𝐶 and the line segment 𝑀𝐵 are radii of circle 𝑀. We can take these four pieces of information and draw some conclusions.

We can say that line segment 𝐴𝐶 is equal to line segment 𝐴𝐵 in length, since this is one of the properties of tangents when they intersect at an external point. We can also say, based on circle properties, that the line segment 𝑀𝐴 bisects the angle 𝐵𝐴𝐶. This is another one of our circle theorems. And since that’s true, angle 𝐵𝐴𝑀 has to be equal in measure to angle 𝑀𝐴𝐶. And angle 𝑀𝐴𝐶 is 36 degrees, which makes angle 𝐵𝐴𝑀 36 degrees. That’s the first part of the question.

Now we’re moving on to find the angle of 𝐴𝑀𝐶. We recognize that the points 𝐴, 𝐶, and 𝑀 form a triangle. And since the line segment 𝐴𝐶 is tangent to this circle at point 𝐶, there’s a right angle here. The measure of angle 𝑀𝐶𝐴 is 90 degrees. And once we see that, we can say that 90 degrees plus 36 degrees plus our missing angle must equal 180 degrees because we know the sum of interior angles in triangles must be 180. 90 plus 36 is 126. And when we subtract 126 degrees from both sides, we see that the measure of angle 𝐴𝑀𝐶 is 54 degrees.