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.