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
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
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
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.