Question Video: Finding the Measure of an Angle Using the Triangles’ and Tangents’ Properties Mathematics • Third Year of Preparatory School

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.