Question Video: Finding the Measure of Two Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations Mathematics • 8th Grade

Given that πβ π· = 2/3 πβ π΅, find πβ π΅ and πβ π·.

02:45

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, weβre going to begin by adding what weβve been told to our figure. Weβll define the measure of angle π΅ as π₯ degrees. So thatβs this one. Weβre 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₯. Thatβ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 π₯ plus 53 plus π₯ plus 127 must be equal to 360 degrees.

Letβs simplify the left-hand side of our equation. Firstly, we add 53 and 127 to get 180. Next, weβll 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 doesnβt actually matter. Letβs 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; itβs 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.