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

Given that π‘šβˆ π‘€π΄πΆ = 36Β°, determine π‘šβˆ π΅π΄π‘€ and π‘šβˆ π΄π‘€πΆ.

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Video Transcript

Given that the measure of angle 𝑀𝐴𝐢 equals 36 degrees, determine the measure of angle 𝐡𝐴𝑀 and the measure of angle 𝐴𝑀𝐢.

We want to know the measure of angle 𝐡𝐴𝑀 and the measure of angle 𝐴𝑀𝐢. Let’s start by identifying what we know. We know that the measure of angle 𝑀𝐴𝐢 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 𝐴𝐢 and the line segment 𝐴𝐡 are tangent to the circle 𝑀 since each segment intersects the circle at exactly one point. Line segment 𝐴𝐢 and line segment 𝐴𝐡 intersect at point 𝐴. From there, we say that line segment 𝑀𝐢 and the line segment 𝑀𝐡 are radii of circle 𝑀. We can take these four pieces of information and draw some conclusions.

We can say that line segment 𝐴𝐢 is equal to line segment 𝐴𝐡 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 𝐡𝐴𝐢. This is another one of our circle theorems. And since that’s true, angle 𝐡𝐴𝑀 has to be equal in measure to angle 𝑀𝐴𝐢. And angle 𝑀𝐴𝐢 is 36 degrees, which makes angle 𝐡𝐴𝑀 36 degrees. That’s the first part of the question.

Now we’re moving on to find the angle of 𝐴𝑀𝐢. We recognize that the points 𝐴, 𝐢, and 𝑀 form a triangle. And since the line segment 𝐴𝐢 is tangent to this circle at point 𝐢, there’s a right angle here. The measure of angle 𝑀𝐢𝐴 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 𝐴𝑀𝐢 is 54 degrees.

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