# Question Video: Determining a Range of Values for the Measure of an Angle Mathematics • 11th Grade

In β³π΄π΅πΆ, the ray π΅π bisects β π΄π΅πΆ, the ray πΆπ bisects β π΄πΆπ΅, and π΄π΅ > π΄πΆ. If πβ ππΆπ΅ = 30Β°, which of the following values can satisfy πβ ππ΅πΆ? [A] 28Β° [B] 30Β° [C] 32Β°

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### Video Transcript

In triangle π΄π΅πΆ, the ray from π΅ through π bisects angle π΄π΅πΆ, the ray from πΆ through π bisects angle π΄πΆπ΅, and π΄π΅ is greater than π΄πΆ. If the measure of angle ππΆπ΅ is equal to 30 degrees, which of the following values can satisfy the measure of angle ππ΅πΆ? 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 π΄π΅πΆ. First, we are told that the ray from π΅ through π bisects the angle at π΅ and the ray from πΆ through π bisects the angle at πΆ. 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 ππΆπ΅ is 30 degrees. We can add this onto our diagram. However, we know that angle ππΆπ΄ has the same measure as angle ππΆπ΅ because the ray from πΆ through π 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 π΄π΅ is longer than the side π΄πΆ. We want to use this information to determine information about the measure of angle ππ΅πΆ. Since we are given a comparison of the lengths of two sides of triangle π΄π΅πΆ, 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 π΄π΅ is longer than π΄πΆ, then the measure of angle πΆ must be greater than the measure of angle π΅. We know that π΄π΅ is longer than π΄πΆ in triangle π΄π΅πΆ. So, we must have that the measure of angle πΆ is greater than the measure of angle π΅.

We can also note that angle πΆ is made up of two angles of measure 30 degrees. In the same way, we can see that angle π΅ is made up of the two congruent angles at π΅. We can say that angle π΅ has measure given by two times the measure of angle ππ΅πΆ. Therefore, the angle comparison theorem in triangles tells us that 60 degrees must be larger than twice the measure of angle ππ΅πΆ. Finally, we can divide the inequality through by two to get that 30 degrees must be greater than the measure of angle ππ΅πΆ. 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.