Given that the line 𝐴𝐵 is a tangent to the circle with center 𝑀 and the measure of
angle 𝑀𝐵𝐹 is 123 degrees, determine the measure of angle 𝐴𝑀𝐵.
Angle 𝐴𝑀𝐵 is the angle formed when we travel from 𝐴 to 𝑀 and then to 𝐵. Angle 𝑀𝐵𝐹 is the angle made when we travel from 𝑀 to 𝐵 to 𝐹, and we’re told
that its measure is 123 degrees. We can see that the angle we’re looking to find, 𝐴𝑀𝐵, 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
First, let’s consider the angle 𝑀𝐵𝐴. 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 𝑀𝐵𝐹 and 𝑀𝐵𝐴 sum to 180 degrees. We can substitute 123 degrees for the measure of angle 𝑀𝐵𝐹 into our equation. And we now have an equation we can solve to find the measure of angle 𝑀𝐵𝐴. 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 𝑀𝐵𝐴 is equal to 57 degrees. So we’ve found one of the angles in triangle 𝑀𝐵𝐴. Can we find another one? What about angle 𝑀𝐴𝐵?
Well, this is the angle formed where a tangent to the circle, that’s the line 𝐴𝐵,
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 𝑀𝐴𝐵 is a right angle, so it has measure 90 degrees.
We’ve therefore found two of the angles in triangle 𝑀𝐵𝐴. 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 𝐴𝑀𝐵 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 𝐴𝑀𝐵 equals 33 degrees.