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

Given that the line 𝐴𝐡 is a tangent to the circle with center 𝑀 and π‘šβˆ π‘€π΅πΉ = 123Β°, π‘šβˆ π΄π‘€π΅.


Video Transcript

Given that the line 𝐴𝐡 is a tangent to the circle with center 𝑀 and the measure of angle 𝑀𝐡𝐹 is 123 degrees, determine the measure of angle 𝐴𝑀𝐡.

Angle 𝐴𝑀𝐡 is the angle formed when we travel from 𝐴 to 𝑀 and then to 𝐡. Angle 𝑀𝐡𝐹 is the angle made when we travel from 𝑀 to 𝐡 to 𝐹, and we’re told that its measure is 123 degrees. We can see that the angle we’re looking to find, 𝐴𝑀𝐡, is contained within a triangle. If we can work out the other two angles in this triangle, we can use the fact that the sum of the angles in any triangle is 180 degrees to find the angle we’re looking for.

First, let’s consider the angle 𝑀𝐡𝐴. One of our most basic angle facts is that the angles on a straight line sum to 180 degrees. And this angle is on a straight line with the angle we’ve already marked as being 123 degrees. So we can say that the measures of angles 𝑀𝐡𝐹 and 𝑀𝐡𝐴 sum to 180 degrees. We can substitute 123 degrees for the measure of angle 𝑀𝐡𝐹 into our equation. And we now have an equation we can solve to find the measure of angle 𝑀𝐡𝐴. To do this, we need to subtract 123 degrees from both sides of the equation. And doing so gives us that the measure of angle 𝑀𝐡𝐴 is equal to 57 degrees. So we’ve found one of the angles in triangle 𝑀𝐡𝐴. Can we find another one? What about angle 𝑀𝐴𝐡?

Well, this is the angle formed where a tangent to the circle, that’s the line 𝐴𝐡, meets the radius of the circle, 𝐴𝑀. And we know that a tangent to a circle is perpendicular to the radius at the point of contact. So we know that angle 𝑀𝐴𝐡 is a right angle, so it has measure 90 degrees.

We’ve therefore found two of the angles in triangle 𝑀𝐡𝐴. And using the angle sum in a triangle, we can find the third. Since the angles in a triangle sum to 180 degrees, we have the equation the measure of angle 𝐴𝑀𝐡 plus 90 degrees plus 57 degrees equals 180 degrees. Subtracting 90 degrees and 57 degrees from both sides then, we have 180 minus 90 minus 57 degrees on our right-hand side. And this is equal to 33 degrees. And so the measure of angle 𝐴𝑀𝐡 equals 33 degrees.

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