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’ll begin by adding the
information in the question to the diagram. The measure of angle 𝑀𝐴𝐶 is
36 degrees. We need to find the measures of
angles 𝐵𝐴𝑀 and 𝐴𝑀𝐶. First, we should know that 𝐴
is a point exterior to the circle. The line segments 𝐴𝐶 and 𝐴𝐵
are tangents of the circle, and the line segment 𝐴𝑀 is the line segment
connecting this exterior point to the center of the circle. We can recall that the line
connecting an exterior point to the center of a circle bisects the angle formed
by two tangents from that point to the circle. So at point 𝐴, the measures of
angles 𝑀𝐴𝐵 and 𝑀𝐴𝐶 are equal. We know the measure of angle
𝑀𝐴𝐶 is 36 degrees, and so the measure of angle 𝑀𝐴𝐵 or angle 𝐵𝐴𝑀 is also
36 degrees.
We’ll now find the measure of
angle 𝐴𝑀𝐶, and to do this, we’ll consider triangle 𝐴𝑀𝐶. 𝐴𝐶 is a tangent of the circle
and 𝑀𝐶 is a radius. We recall that any tangent to a
circle is perpendicular to the radius at the point of contact, and so the angle
𝑀𝐶𝐴 is a right angle. Using the fact that the angle
sum in any triangle is 180 degrees, we find the measure of angle 𝐴𝑀𝐶 by
subtracting the measures of the other two angles in triangle 𝐴𝑀𝐶 from 180
degrees, which gives 54 degrees. The measure of angle 𝐵𝐴𝑀 is
36 degrees, and the measure of angle 𝐴𝑀𝐶 is 54 degrees.
