# Question Video: Observing Relationships between Multiple Bisected Angles Mathematics

Draw ray π΅π΄ and ray π΅πΆ to create an obtuse angle π΄π΅πΆ. Draw ray π΅π· to bisect β π΄π΅πΆ; then draw ray π΅πΈ to bisect β π·π΅πΆ and ray π΅πΉ to bisect β π΄π΅π·. From your constructions, which of the following is true? [A] πβ π΄π΅πΉ = 2 πβ πΆπ΅πΈ [B] πβ πΆπ΅πΈ = πβ π·π΅π΄ [C] πβ πΈπ΅π· = (1/2) πβ π·π΅πΉ [D] πβ πΆπ΅π· = 2 πβ πΉπ΅π΄

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### Video Transcript

Draw ray π΅π΄ and ray π΅πΆ to create an obtuse angle π΄π΅πΆ. Draw ray π΅π· to bisect angle π΄π΅πΆ; then draw ray π΅πΈ to bisect angle π·π΅πΆ and ray π΅πΉ to bisect angle π΄π΅π·. From your constructions, which of the following is true? Option (A) the measure of angle π΄π΅πΉ is equal to two times the measure of angle πΆπ΅πΈ. Option (B) the measure of angle πΆπ΅πΈ is equal to the measure of angle π·π΅π΄. Option (C) the measure of angle πΈπ΅π· is equal to one-half times the measure of angle π·π΅πΉ. Or option (D) the measure of angle πΆπ΅π· is equal to two times the measure of angle πΉπ΅π΄.

Weβll begin by drawing an obtuse angle π΄π΅πΆ 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΅πΆ, as shown. This forms an obtuse angle π΄π΅πΆ. Next, we will draw ray π΅π· to bisect angle π΄π΅πΆ. 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΅πΆ. 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΅πΆ. According to the definition of an angle bisector, we note that angle πΆπ΅π· is congruent to angle π·π΅π΄.

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

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

Weβve now bisected each of the angles created by the original angle bisector. Letβs examine our construction to see if we find any other congruencies. Letβs say that the measure of angle πΉπ΅π΄ is π₯ degrees. Because it is congruent to angle π·π΅πΉ, we have that the measure of angle π·π΅πΉ 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 πΆπ΅π·, we also know that the measure of angle πΆπ΅π· equals two π₯ degrees. Then, because angle πΆπ΅π· is bisected by ray π΅πΈ, we know it is split in half. And half of two π₯ is π₯. Therefore, both angle πΆπ΅πΈ and angle πΈπ΅π· measure π₯ degrees.

So, by applying the definition of an angle bisector to each construction, we have shown that angles πΉπ΅π΄, π·π΅πΉ, πΈπ΅π·, and πΆπ΅πΈ all have equal measure. Now, we are ready to consider the given answers to determine which one is true.

Letβs start with answer (A). Is the measure of angle π΄π΅πΉ equal to twice the measure of angle πΆπ΅πΈ? As shown in the diagram, we have the measure of angle π΄π΅πΉ, which is π₯ degrees, and the measure of angle πΆπ΅πΈ, 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 πΆπ΅πΈ 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 πΆπ΅πΈ. So answer (B) is a false statement.

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

Answer (D) claims that the measure of angle πΆπ΅π· is equal to two times the measure of angle πΉπ΅π΄. We know that the measure of angle πΆπ΅π· is two π₯ degrees and the measure of angle πΉπ΅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.