Video Transcript
Given that line 𝐴𝐵 is a tangent
to the circle with center 𝑀 at 𝐴 and the measure of angle 𝑀𝐵𝐸 equals 151
degrees, find the measure of angle 𝐴𝑀𝐵.
In this problem, we are given a
circle whose center is 𝑀. And we are given the information
that line 𝐴𝐵 is a tangent to the circle. We can recall that any tangent to a
circle is perpendicular to the radius at the point of contact. The point of contact between the
radius, which is the line segment 𝐴𝑀, and the tangent line 𝐴𝐵 is at point
𝐴. So the measure of angle 𝑀𝐴𝐵 must
be 90 degrees.
Now, we are asked to find the
measure of angle 𝐴𝑀𝐵. We can do this by using the
triangle 𝐴𝐵𝑀. Observe that we can calculate the
measure of angle 𝐴𝐵𝑀 since we have a straight line 𝐴𝐵 and the sum of the angle
measures on a straight line sum to 180 degrees. So the measure of angle 𝐴𝐵𝑀 is
equal to 180 degrees minus 151 degrees, which is 29 degrees. And now we have two angles in the
triangle 𝐴𝐵𝑀, we can find the third angle.
We know that the internal angle
measures in a triangle sum to 180 degrees. So we can write that 29 degrees
plus 90 degrees plus the measure of angle 𝐴𝑀𝐵 is equal to 180 degrees. Simplifying, we have that 119
degrees plus the measure of angle 𝐴𝑀𝐵 equals 180 degrees. And then subtracting 119 degrees
from both sides, we have the answer that the measure of angle 𝐴𝑀𝐵 is 61
degrees.
