Given that the measure of angle 𝐷
is two-thirds the measure of angle 𝐵, find the measure of angle 𝐵 and the measure
of angle 𝐷.
To answer this question, we’re
going to begin by adding what we’ve been told to our figure. We’ll define the measure of angle
𝐵 as 𝑥 degrees. So that’s this one. We’re told that the measure of
angle 𝐷 is two-thirds the measure of angle 𝐵. So the measure of angle 𝐷 must be
two-thirds of 𝑥. That’s this one. Okay, so how does that help us? Well, we know that the interior
angles in a quadrilateral, a four-sided polygon, add to make 360 degrees. We can therefore say that
two-thirds 𝑥 plus 53 plus 𝑥 plus 127 must be equal to 360 degrees.
Let’s simplify the left-hand side
of our equation. Firstly, we add 53 and 127 to get
180. Next, we’ll add two-thirds 𝑥 and
𝑥. And we do so by writing 𝑥 as 𝑥
over one and then creating a common denominator of three by multiplying the
numerator and denominator by three. So we get two-thirds 𝑥 plus
three-thirds 𝑥, which is five-thirds 𝑥. So our equation is five-thirds 𝑥
plus 180 equals 360. We solve for 𝑥 by performing a
series of inverse operations.
First, we subtract 180 from both
sides. That leaves us with five-thirds 𝑥
on the left-hand side and 180 on the right. Now, the next thing we could do is
divide through by five-thirds. Alternatively, we can perform this
in two separate steps, the order of which doesn’t actually matter. Let’s multiply both sides of the
equation by three. On the left-hand side, that leaves
us with simply five 𝑥. And on the right-hand side, we get
540. Next, we divide through by five,
giving us 𝑥 on the left-hand side and 108 on the right.
So we found 𝑥 and therefore the
measure of the angle at 𝐵 to be 𝑥 degrees. But what about the measure of the
angle at 𝐷? Remember, we said that that was
two-thirds 𝑥. So if the measure of the angle at
𝐵 is 108, the measure of the angle at 𝐷 is two-thirds of 108 or two-thirds times
108. Well, one-third of 108 is 36, so
two-thirds is double this; it’s 72. And so we find the measure of the
angle at 𝐵 is 108 degrees and the measure of the angle at 𝐷 is 72 degrees. Remember, we could of course check
by adding 108, 72, 53, and 127 and making sure we do indeed get 360 degrees.