Question Video: Observing Relationships between Multiple Bisected Angles Mathematics

Video Transcript

Draw ray 𝐵𝐴 and ray 𝐵𝐶 to create an obtuse angle 𝐴𝐵𝐶. Draw ray 𝐵𝐷 to bisect angle 𝐴𝐵𝐶; then draw ray 𝐵𝐸 to bisect angle 𝐷𝐵𝐶 and ray 𝐵𝐹 to bisect angle 𝐴𝐵𝐷. From your constructions, which of the following is true? Option (A) the measure of angle 𝐴𝐵𝐹 is equal to two times the measure of angle 𝐶𝐵𝐸. Option (B) the measure of angle 𝐶𝐵𝐸 is equal to the measure of angle 𝐷𝐵𝐴. Option (C) the measure of angle 𝐸𝐵𝐷 is equal to one-half times the measure of angle 𝐷𝐵𝐹. Or option (D) the measure of angle 𝐶𝐵𝐷 is equal to two times the measure of angle 𝐹𝐵𝐴.

We’ll begin by drawing an obtuse angle 𝐴𝐵𝐶 as described. We recall that an obtuse angle will measure more than 90 degrees but less than 180 degrees. Using a straight edge, we can sketch ray 𝐵𝐴. Then, we draw ray 𝐵𝐶, as shown. This forms an obtuse angle 𝐴𝐵𝐶. Next, we will draw ray 𝐵𝐷 to bisect angle 𝐴𝐵𝐶. We recall that to bisect an angle is to split it into two congruent angles.

Now, we must recall the steps to construct an angle bisector. We will need a compass and straightedge or ruler for this process. The first step is to use a compass to trace a circle centered at the vertex that intersects the sides of the angle. In this case, our circle is centered at 𝐵 and intersects both ray 𝐵𝐴 and ray 𝐵𝐶. The next step also requires a compass.

Next, we trace two circles of the same radius centered at the two points we found in the first step. We may need to increase the radius setting on our compass to ensure these circles will intersect on the interior of this obtuse angle. As directed, we name the intersection 𝐷. Then, we use a straightedge to sketch the ray from point 𝐵 through point 𝐷. This is the bisector of angle 𝐴𝐵𝐶. According to the definition of an angle bisector, we note that angle 𝐶𝐵𝐷 is congruent to angle 𝐷𝐵𝐴.

Next, we will draw ray 𝐵𝐸 bisecting angle 𝐷𝐵𝐶. Following the process used in our first construction, we use a compass and straightedge to draw the bisector of angle 𝐷𝐵𝐶. As shown, ray 𝐵𝐸 now bisects angle 𝐷𝐵𝐶. We stop to note that angle 𝐶𝐵𝐷 has been split in half, so the angle 𝐶𝐵𝐸 is congruent to the angle 𝐸𝐵𝐷.

Finally, we follow the same process to bisect angle 𝐴𝐵𝐷 with ray 𝐵𝐹, which looks like this. Ray 𝐵𝐹 is the bisector of angle 𝐴𝐵𝐷. Therefore, angle 𝐹𝐵𝐴 is congruent to angle 𝐷𝐵𝐹.

We’ve now bisected each of the angles created by the original angle bisector. Let’s examine our construction to see if we find any other congruencies. Let’s say that the measure of angle 𝐹𝐵𝐴 is 𝑥 degrees. Because it is congruent to angle 𝐷𝐵𝐹, we have that the measure of angle 𝐷𝐵𝐹 is also equal to 𝑥 degrees. By adding these adjacent angles together, we get the measure of angle 𝐷𝐵𝐴, which is two 𝑥 degrees. Then, because angle 𝐷𝐵𝐴 is congruent to angle 𝐶𝐵𝐷, we also know that the measure of angle 𝐶𝐵𝐷 equals two 𝑥 degrees. Then, because angle 𝐶𝐵𝐷 is bisected by ray 𝐵𝐸, we know it is split in half. And half of two 𝑥 is 𝑥. Therefore, both angle 𝐶𝐵𝐸 and angle 𝐸𝐵𝐷 measure 𝑥 degrees.

So, by applying the definition of an angle bisector to each construction, we have shown that angles 𝐹𝐵𝐴, 𝐷𝐵𝐹, 𝐸𝐵𝐷, and 𝐶𝐵𝐸 all have equal measure. Now, we are ready to consider the given answers to determine which one is true.

Let’s start with answer (A). Is the measure of angle 𝐴𝐵𝐹 equal to twice the measure of angle 𝐶𝐵𝐸? As shown in the diagram, we have the measure of angle 𝐴𝐵𝐹, which is 𝑥 degrees, and the measure of angle 𝐶𝐵𝐸, which is also 𝑥 degrees. However, substituting these values into answer (A) does not balance the equation. We know that for a positive value 𝑥, 𝑥 cannot equal two times 𝑥. So answer (A) is a false statement.

Answer (B) claims that the measure of angle 𝐶𝐵𝐸 is equal to the measure of angle 𝐷𝐵𝐴. But we know that 𝐷𝐵𝐴 measures two 𝑥 degrees, which does not equal the measure of angle 𝐶𝐵𝐸. So answer (B) is a false statement.

Answer (C) claims that the measure of angle 𝐸𝐵𝐷 is equal to half of the measure of angle 𝐷𝐵𝐹. We know that both angle 𝐸𝐵𝐷 and angle 𝐷𝐵𝐹 measure 𝑥 degrees. So this statement is false.

Answer (D) claims that the measure of angle 𝐶𝐵𝐷 is equal to two times the measure of angle 𝐹𝐵𝐴. We know that the measure of angle 𝐶𝐵𝐷 is two 𝑥 degrees and the measure of angle 𝐹𝐵𝐴 is 𝑥 degrees. This time, substituting these expressions into answer (D) balances the equation. According to our constructions, we conclude that answer (D) is the true statement.