Video: Finding the Measures of Angles in a Quadrilateral given a Relation between Them by Solving Linear Equations

Video Transcript

From the figure, in which the measure of angle 𝐶𝐷𝐴 is five 𝑥 degrees, the measure of angle 𝐵𝐶𝐷 is seven 𝑦 degrees, and the measure of angle 𝐴𝐵𝐶 is eight 𝑦 degrees, find the values of 𝑥 and 𝑦.

Let’s begin by adding what we’ve been told to our figure. The measure of angle 𝐶𝐷𝐴 is five 𝑥; that’s this one. The angle 𝐵𝐶𝐷 is seven 𝑦; that’s this one. And the measure of angle 𝐴𝐵𝐶, that’s this one, is eight 𝑦. Okay, so how does this help us? Well, we have a couple of useful facts. Firstly, we know that the sum of the interior angles in a quadrilateral, that’s a four-sided polygon, is 360 degrees. So according to our diagram, we can say that eight 𝑦 plus seven 𝑦 plus five 𝑥 plus 85 equals 360. Let’s add seven 𝑦 and eight 𝑦 to get 15𝑦.

Then we spot that we have two parallel lines in our diagram. The parallel lines are 𝐵𝐴 and 𝐶𝐷. We can quote that cointerior angles sum to 180 degrees. So these two angles, 85 and 5𝑥, are cointerior. They add to 180. And we form a second equation; this time, 85 plus five 𝑥 equals 180. Let’s solve this equation for 𝑥. We subtract 85 from both sides, so five 𝑥 is equal to 95. Then we divide through by five. So 𝑥 is 95 divided by five, which is 19. Remember, we still need to calculate the value of 𝑦. So let’s clear some space.

We’re going to go back to the first equation we formed and substitute 𝑥 equals 19. When we do, we get 15𝑦 plus five times 19 plus 85 equals 360. That’s 15𝑦 plus 95 plus 85 equals 360. 95 plus 85 is 180. So we have an equation in 𝑦 that we can solve. We subtract 180 from both sides. So 15𝑦 is 180. And then, we divide through by 15. 𝑦 is 180 divided by 15, which is equal to 12. So 𝑥 is 19 and 𝑦 is equal to 12.

Now, with angle questions, there’s often more than one way to answer them. We could have used the fact that the angles at 𝐵 and 𝐶 are cointerior. They sum to 180 degrees. We form an equation, and we get eight 𝑦 plus seven 𝑦 equals 180. So 15𝑦 is equal to 180. Once again, we solve this equation by dividing by 15. And we get 𝑦 is equal to 12.