Video Transcript
Two straight lines, 𝐴𝐵 and
𝐶𝐷, intersect at point 𝐸. Fill in the blank. If the angles 𝐴𝐸𝐷 and 𝐴𝐸𝐶
are adjacent angles, where the union of rays 𝐸𝐶 and 𝐸𝐷 equals the line
segment 𝐶𝐷, then the measure of angle 𝐴𝐸𝐶 plus the measure of angle 𝐴𝐸𝐷
equals what. Fill in the blank. If the angles 𝐴𝐸𝐶 and 𝐶𝐸𝐵
are adjacent angles, where the union of rays 𝐸𝐴 and 𝐸𝐵 equals the line
segment 𝐴𝐵, then the measure of angle 𝐴𝐸𝐶 plus the measure of angle 𝐶𝐸𝐵
equals what. True or False: We deduce from
the two parts above that the measure of angle 𝐴𝐸𝐷 equals the measure of angle
𝐶𝐸𝐵.
Let’s begin this question by
drawing the two given lines, 𝐴𝐵 and 𝐶𝐷, which intersect at a point 𝐸. Notice that we could have drawn
any different diagram of the lines 𝐴𝐵 and 𝐶𝐷 that intersect at point 𝐸, so
long as it shows that important line and intersection information. We would still be able to use
any such diagram to answer the questions.
So let’s use the first diagram
and look at the first part of this question. Here, we need to first identify
the angles 𝐴𝐸𝐷 and 𝐴𝐸𝐶. The second part of this
sentence, which tells us that the union of rays 𝐸𝐶 and 𝐸𝐷 is the line
segment 𝐶𝐷, is really stating the fact that these lines form one straight-line
segment. And what do we know about the
angles on a straight line? Well, the angle measures on a
straight line sum to 180 degrees. And so, the measure of angle
𝐴𝐸𝐷 plus the measure of angle 𝐴𝐸𝐶 is equal to 180 degrees. And that’s the first missing
blank completed.
Let’s look at the second part
of the question. This time, we’re looking at the
angles 𝐴𝐸𝐶 and 𝐶𝐸𝐵. Once again, we’re told that the
rays 𝐸𝐴 and 𝐸𝐵 form one straight-line segment, 𝐴𝐵. And we know that the angle
measures on a straight line sum to 180 degrees. So these two angle measures of
𝐴𝐸𝐶 and 𝐶𝐸𝐵 will also sum to 180 degrees. So now we have answered the
second part of this question.
Let’s look at the final
part. In this part, we are
considering the angle measures of 𝐴𝐸𝐷 and 𝐶𝐸𝐵. To help us with this, we’ll
consider what we discovered in parts one and two. In the first part, we
recognized that the measures of angles 𝐴𝐸𝐶 and 𝐴𝐸𝐷 added to give 180
degrees. Let’s label the measure of
angle 𝐴𝐸𝐷 as 𝑥 degrees and the measure of angle 𝐴𝐸𝐶 as 𝑦 degrees. In the second part of the
question, we identified another pair of angle measures that added to 180
degrees. And since 𝑥 degrees plus 𝑦
degrees equals 180 degrees, then we can say that the measure of angle 𝐶𝐸𝐵 is
also 𝑥 degrees.
So, we can say that the
statement that the measure of angle 𝐴𝐸𝐷 equals the measure of angle 𝐶𝐸𝐵 is
true. And in fact, what we have here
is a proof that vertically opposite angles are equal. We could even have continued in
this example to demonstrate that the measure of angle 𝐴𝐸𝐶 equals the measure
of angle 𝐷𝐸𝐵. These vertically opposite
angles will both be 𝑦 degrees.
