Question Video: Finding the Measure of an Inscribed Angle by Using the Properties of Tangents to the Circle Mathematics

Video Transcript

Given that line 𝐴𝐵 is a tangent to a circle with center 𝑀 and the measure of angle 𝐴𝐵𝑀 is equal to 49 degrees, determine the measure of angle 𝐴𝐷𝐵.

Angle 𝐴𝐷𝐵 is the angle formed when we travel from 𝐴 to 𝐷 to 𝐵, so it’s the angle marked in orange in the diagram. Angle 𝐴𝐵𝑀 is the angle formed when we travel from 𝐴 to 𝐵 to 𝑀. It’s the angle now marked in pink on the diagram with a measure of 49 degrees.

From the information given, we aren’t able to calculate the measure of angle 𝐴𝐷𝐵 directly. We’re going to need to find the measures of some other angles in the figure first. The other key piece of information given in the question is that line 𝐴𝐵 is a tangent to the circle with center 𝑀. And a key property about tangents of circles is that a tangent to a circle is perpendicular to the radius at the point of contact.

The point where the tangent meets the circle is point 𝐴, and the radius here is the line segment 𝐴𝑀. So, we know that the measure of angle 𝐵𝐴𝑀 is 90 degrees. Now we know one more angle within the figure. However, we still aren’t able to calculate the measure of angle 𝐴𝐷𝐵 directly, so let’s see what other angles we can work out.

We have a triangle. In fact, triangle 𝐴𝑀𝐵 is a right triangle, and we know two of its angles, the right angle and the angle measuring 49 degrees. So, using the fact that the angle measures in a triangle sum to 180 degrees, we can calculate the third angle in this triangle. We do this by writing an equation which states that the measure of angle 𝐴𝑀𝐵 plus 49 degrees plus 90 degrees is equal to 180 degrees. 49 plus 90 is 139, and subtracting this from 180, we find the measure of angle 𝐴𝑀𝐵 is 41 degrees. So we now know another angle in our diagram.

We still don’t have enough information to calculate the measure of an angle 𝐴𝐷𝐵, but we can now calculate a different angle, angle 𝐴𝑀𝐷. We know that the angle measures on a straight line sum to 180 degrees. So, the measure of angle 𝐴𝑀𝐷 plus the angle measure we’ve just calculated, 41 degrees, must equal 180 degrees. The measure of angle 𝐴𝑀𝐷 is therefore equal to 180 degrees minus 41 degrees. That’s 139 degrees. Now we found almost all the angles in the figure, but still not the one we were looking for.

The final step is to consider triangle 𝐴𝑀𝐷, in which we know one angle is 139 degrees. We recognize both line segment 𝑀𝐴 and line segment 𝑀𝐷 as radii of the same circle with center 𝑀. Therefore, they have the same length. It follows that triangle 𝑀𝐷𝐴 is an isosceles triangle, and this means that angles 𝐷𝐴𝑀 and 𝐴𝐷𝑀 have equal measure. We can therefore find the measure of each angle by subtracting the measure of the third angle, 139 degrees, from the total angle sum in a triangle, 180 degrees, and then splitting the remainder in half. Doing so gives each of these angles a measure of 20.5 degrees.

Now we know that angle 𝐴𝐷𝑀 is in fact the same as angle 𝐴𝐷𝐵. They both refer to the angle highlighted here in pink. And so, we’ve completed the problem. By using some of the more basic facts of angles in triangles and angles in straight lines and the key property that the tangent to a circle is perpendicular to the radius at the point of contact, we found that the measure of angle 𝐴𝐷𝐵 is 20.5 degrees.