### Video Transcript

Draw ray π΅π΄ and ray π΅πΆ to
create an obtuse angle π΄π΅πΆ. Draw ray π΅π· to bisect angle
π΄π΅πΆ; then draw ray π΅πΈ to bisect angle π·π΅πΆ and ray π΅πΉ to bisect angle
π΄π΅π·. From your constructions, which of
the following is true? Option (A) the measure of angle
π΄π΅πΉ is equal to two times the measure of angle πΆπ΅πΈ. Option (B) the measure of angle
πΆπ΅πΈ is equal to the measure of angle π·π΅π΄. Option (C) the measure of angle
πΈπ΅π· is equal to one-half times the measure of angle π·π΅πΉ. Or option (D) the measure of angle
πΆπ΅π· is equal to two times the measure of angle πΉπ΅π΄.

Weβll begin by drawing an obtuse
angle π΄π΅πΆ 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 π΅π΄. Then, we draw ray π΅πΆ, as
shown. This forms an obtuse angle
π΄π΅πΆ. Next, we will draw ray π΅π· to
bisect angle π΄π΅πΆ. 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 π΅ and intersects both ray π΅π΄ and ray π΅πΆ. 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 π·. Then, we use a straightedge to
sketch the ray from point π΅ through point π·. This is the bisector of angle
π΄π΅πΆ. According to the definition of an
angle bisector, we note that angle πΆπ΅π· is congruent to angle π·π΅π΄.

Next, we will draw ray π΅πΈ
bisecting angle π·π΅πΆ. Following the process used in our
first construction, we use a compass and straightedge to draw the bisector of angle
π·π΅πΆ. As shown, ray π΅πΈ now bisects
angle π·π΅πΆ. We stop to note that angle πΆπ΅π·
has been split in half, so the angle πΆπ΅πΈ is congruent to the angle πΈπ΅π·.

Finally, we follow the same process
to bisect angle π΄π΅π· with ray π΅πΉ, which looks like this. Ray π΅πΉ is the bisector of angle
π΄π΅π·. Therefore, angle πΉπ΅π΄ is
congruent to angle π·π΅πΉ.

Weβve now bisected each of the
angles created by the original angle bisector. Letβs examine our construction to
see if we find any other congruencies. Letβs say that the measure of angle
πΉπ΅π΄ is π₯ degrees. Because it is congruent to angle
π·π΅πΉ, we have that the measure of angle π·π΅πΉ is also equal to π₯ degrees. By adding these adjacent angles
together, we get the measure of angle π·π΅π΄, which is two π₯ degrees. Then, because angle π·π΅π΄ is
congruent to angle πΆπ΅π·, we also know that the measure of angle πΆπ΅π· equals two
π₯ degrees. Then, because angle πΆπ΅π· is
bisected by ray π΅πΈ, we know it is split in half. And half of two π₯ is π₯. Therefore, both angle πΆπ΅πΈ and
angle πΈπ΅π· measure π₯ degrees.

So, by applying the definition of
an angle bisector to each construction, we have shown that angles πΉπ΅π΄, π·π΅πΉ,
πΈπ΅π·, and πΆπ΅πΈ all have equal measure. Now, we are ready to consider the
given answers to determine which one is true.

Letβs start with answer (A). Is the measure of angle π΄π΅πΉ
equal to twice the measure of angle πΆπ΅πΈ? As shown in the diagram, we have
the measure of angle π΄π΅πΉ, which is π₯ degrees, and the measure of angle πΆπ΅πΈ,
which is also π₯ degrees. However, substituting these values
into answer (A) does not balance the equation. We know that for a positive value
π₯, π₯ cannot equal two times π₯. So answer (A) is a false
statement.

Answer (B) claims that the measure
of angle πΆπ΅πΈ is equal to the measure of angle π·π΅π΄. But we know that π·π΅π΄ measures
two π₯ degrees, which does not equal the measure of angle πΆπ΅πΈ. So answer (B) is a false
statement.

Answer (C) claims that the measure
of angle πΈπ΅π· is equal to half of the measure of angle π·π΅πΉ. We know that both angle πΈπ΅π· and
angle π·π΅πΉ measure π₯ degrees. So this statement is false.

Answer (D) claims that the measure
of angle πΆπ΅π· is equal to two times the measure of angle πΉπ΅π΄. We know that the measure of angle
πΆπ΅π· is two π₯ degrees and the measure of angle πΉπ΅π΄ is π₯ 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.