Video Transcript
Find the measure of angle 𝐷.
We can identify on the diagram that
there are two pairs of parallel line segments marked, with the first being that line
segment 𝐴𝐵 is parallel to line segment 𝐷𝐶. And secondly, line segments 𝐴𝐷
and 𝐵𝐹 are also marked as parallel. Both of these pairs of parallel
lines may be useful in helping us work out some unknown angle measures. The angle 𝐷 whose measure we need
to determine is in the quadrilateral 𝐴𝐵𝐶𝐷. But even if we use the information
that this is a quadrilateral to help us calculate this angle measure, at the minute
we only have the measure of one other angle in this polygon. So let’s see what other angle
measures we can determine.
We can return to these parallel
line segments 𝐷𝐶 and 𝐴𝐵. These parallel lines are cut by the
transversal of line segment 𝐵𝐶. And we can recall that there is a
relationship between these interior angles at vertices 𝐶 and 𝐵. It is that if a transversal cuts a
pair of parallel lines, then the interior angles on the same side of the transversal
are supplementary. And that’s what we have here. We can write that the two angle
measures at vertices 𝐶 and 𝐵 must sum to 180 degrees. And we know that the angle measure
at 𝐶 is 124 degrees. So, subtracting 124 degrees from
180 degrees, we have that the measure of angle 𝐵 is 56 degrees.
But this still isn’t enough to help
us find the measure of angle 𝐷. We would also need to know the
measure of angle 𝐴 to do this. Now, angle 𝐴 is formed between one
of a pair of parallel lines and a transversal of line 𝐴𝐵. That means that we can identify
that angles 𝐷𝐴𝐸 and 𝐹𝐵𝐸 are corresponding angles. And corresponding angles in a
transversal of parallel lines are congruent. So the measure of angle 𝐷𝐴𝐸 is
equal to the measure of angle 𝐹𝐵𝐸.
But what is the measure of angle
𝐹𝐵𝐸? Well, we are given a clue to help
us, in that these two angles 𝐶𝐵𝐹 and 𝐹𝐵𝐸 are marked as congruent. So if we said that angle 𝐹𝐵𝐸 is
𝑥 degrees, then angle 𝐶𝐵𝐹 would also be 𝑥 degrees. And indeed, because we have found
these corresponding angles, then angle 𝐷𝐴𝐸 is 𝑥 degrees too. The clue to working out the value
of 𝑥 is that these three angles created at vertex 𝐵 lie on a straight line and the
sum of the measures of the angles on a straight line is 180 degrees. Therefore, adding 56 degrees, 𝑥
degrees, and 𝑥 degrees would equal 180 degrees. By simplifying, we have that two 𝑥
degrees equals 124 degrees. So 𝑥 equals 62, and we have three
congruent angles of measure 62 degrees.
Now, we can use the quadrilateral
𝐴𝐵𝐶𝐷 with the three known angle measures to help us work out the fourth unknown
angle measure of angle 𝐷. Given that the interior angles in a
quadrilateral sum to 360 degrees, we can write that the four angles of 62 degrees
plus 56 degrees plus 124 degrees plus the measure of angle 𝐷 equals 360
degrees. Simplifying the left-hand side, we
have that 242 degrees plus the measure of angle 𝐷 equals 360 degrees. And subtracting 242 degrees from
both sides of this equation, we have found the answer that the measure of angle 𝐷
is 118 degrees.
