Question Video: Finding the Size of an Angle in a Circle Using Colloralies of Parallel Chords Mathematics

Video Transcript

Given that the measure of angle 𝑀𝐴𝐶 equals 48 degrees, find 𝐿.

Let’s identify what we know based on this image. The figure tells us that line segment 𝐴𝐶 is parallel to line segment 𝑀𝐵. Since these two lines are parallel, this angle 𝑀𝐴𝐶 and this angle 𝐴𝑀𝐵 are supplementary angles. The two of them when added together equal 180 degrees.

We know that 𝑀𝐴𝐶 is 48 degrees. If we solve this statement, we can find the measure of angle 𝐴𝑀𝐵. Subtract 48 degrees from both sides, and we find out that the measure of angle 𝐴𝑀𝐵 is 132 degrees. This is not enough information to solve. We also need to know this: the measure of an inscribed angle.

For us, the angle 𝐵𝐶𝐴 is half the measure of the central angle subtended by the same arc. Which angle is that? That would be this angle. The measure of angle 𝐵𝐶𝐴 is equal to one-half of the central angle. But how will we find out what the central angle is?

We know that a circle is 360 degrees, and we know that the measure of angle 𝐴𝑀𝐵 is 132 degrees. 360 minus 132 equals 228 degrees. The central angle subtended by the arc 𝐵𝐶𝐴 equals 228 degrees. One-half of 228 equals 114 degrees, and that means 𝐿 equals 114 degrees.