### Video Transcript

Given that the measure of angle π·
is two-thirds the measure of angle π΅, find the measure of angle π΅ and the measure
of angle π·.

To answer this question, weβre
going to begin by adding what weβve been told to our figure. Weβll define the measure of angle
π΅ as π₯ degrees. So thatβs this one. Weβre told that the measure of
angle π· is two-thirds the measure of angle π΅. So the measure of angle π· must be
two-thirds of π₯. Thatβs 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 π₯ plus 53 plus π₯ plus 127 must be equal to 360 degrees.

Letβs simplify the left-hand side
of our equation. Firstly, we add 53 and 127 to get
180. Next, weβll add two-thirds π₯ and
π₯. And we do so by writing π₯ as π₯
over one and then creating a common denominator of three by multiplying the
numerator and denominator by three. So we get two-thirds π₯ plus
three-thirds π₯, which is five-thirds π₯. So our equation is five-thirds π₯
plus 180 equals 360. We solve for π₯ by performing a
series of inverse operations.

First, we subtract 180 from both
sides. That leaves us with five-thirds π₯
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 doesnβt actually matter. Letβs multiply both sides of the
equation by three. On the left-hand side, that leaves
us with simply five π₯. And on the right-hand side, we get
540. Next, we divide through by five,
giving us π₯ on the left-hand side and 108 on the right.

So we found π₯ and therefore the
measure of the angle at π΅ to be π₯ degrees. But what about the measure of the
angle at π·? Remember, we said that that was
two-thirds π₯. So if the measure of the angle at
π΅ is 108, the measure of the angle at π· is two-thirds of 108 or two-thirds times
108. Well, one-third of 108 is 36, so
two-thirds is double this; itβs 72. And so we find the measure of the
angle at π΅ is 108 degrees and the measure of the angle at π· 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.