Video Transcript
Two circles with centers 𝑀 and 𝑁 are tangent at 𝐴. Given that the measure of angle 𝐴𝐵𝑀 is 58 degrees, find the measure of angle 𝐵𝑀𝑁.
In this question, we have two circles that have a common tangent at point 𝐴. This means that they touch internally at this point. We recall from our properties of circles that the tangent to a circle is perpendicular to the radius at the point of contact. This means that the measure of angle 𝑀𝐴𝐵 is 90 degrees. And the triangle 𝑀𝐴𝐵 is a right triangle.
We are told in the question that the measure of angle 𝐴𝐵𝑀 is 58 degrees. And we need to calculate the measure of angle 𝐵𝑀𝑁, which is the same as angle 𝐵𝑀𝐴 in our triangle. We recall that angles in a triangle sum to 180 degrees. This means that the measure of angle 𝐵𝑀𝐴 plus 58 degrees plus 90 degrees is equal to 180 degrees. 58 degrees plus 90 degrees is equal to 148 degrees. We can then subtract this from both sides of our equation such that the measure of angle 𝐵𝑀𝐴 is 32 degrees.
As already mentioned, this has the same measure as angle 𝐵𝑀𝑁. The correct answer is 32 degrees.
