In triangle 𝐴𝐵𝐶, the ray from 𝐵 through 𝑀 bisects angle 𝐴𝐵𝐶, the ray from 𝐶 through 𝑀 bisects angle 𝐴𝐶𝐵, and 𝐴𝐵 is greater than 𝐴𝐶. If the measure of angle 𝑀𝐶𝐵 is equal to 30 degrees, which of the following values can satisfy the measure of angle 𝑀𝐵𝐶? Option (A) 28 degrees. Option (B) 30 degrees. Or is it option (C) 32 degrees?
In this question, we are given some information about a triangle 𝐴𝐵𝐶. First, we are told that the ray from 𝐵 through 𝑀 bisects the angle at 𝐵 and the ray from 𝐶 through 𝑀 bisects the angle at 𝐶. Remember, an angle bisector will bisect the angle into two angles of equal measure. This information is already included in the given diagram. However, we can also highlight this information as shown.
We are also told that the measure of angle 𝑀𝐶𝐵 is 30 degrees. We can add this onto our diagram. However, we know that angle 𝑀𝐶𝐴 has the same measure as angle 𝑀𝐶𝐵 because the ray from 𝐶 through 𝑀 is the angle bisector. So, we can also add that this angle has measure 30 degrees onto the diagram.
The last piece of information we are told is that the side 𝐴𝐵 is longer than the side 𝐴𝐶. We want to use this information to determine information about the measure of angle 𝑀𝐵𝐶. Since we are given a comparison of the lengths of two sides of triangle 𝐴𝐵𝐶, we can compare the measures of the angles opposite these sides using the angle comparison theorem in triangles. This tells us that the angle opposite the longer side will have the larger measure. More formally, we can recall that if 𝐴𝐵 is longer than 𝐴𝐶, then the measure of angle 𝐶 must be greater than the measure of angle 𝐵. We know that 𝐴𝐵 is longer than 𝐴𝐶 in triangle 𝐴𝐵𝐶. So, we must have that the measure of angle 𝐶 is greater than the measure of angle 𝐵.
We can also note that angle 𝐶 is made up of two angles of measure 30 degrees. In the same way, we can see that angle 𝐵 is made up of the two congruent angles at 𝐵. We can say that angle 𝐵 has measure given by two times the measure of angle 𝑀𝐵𝐶. Therefore, the angle comparison theorem in triangles tells us that 60 degrees must be larger than twice the measure of angle 𝑀𝐵𝐶. Finally, we can divide the inequality through by two to get that 30 degrees must be greater than the measure of angle 𝑀𝐵𝐶. Hence, its measure must be smaller than 30 degrees. We see that only option (A), 28 degrees, is less than 30 degrees. So, it is the only possible correct answer.