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Lesson Video: The Sum of Angles in Quadrilaterals Mathematics • 8th Grade

In this video, we will learn how to find a missing angle’s measure in a quadrilateral knowing that the sum of angles in a quadrilateral is 360 degrees.

15:44

Video Transcript

In this video, we’ll learn how to find a missing angle’s measure in a quadrilateral, knowing that the sum of angles in a quadrilateral is 360 degrees. We’ll begin by looking at a short geometrical proof of this fact before considering how it can help us to solve problems involving missing angles in quadrilaterals.

We begin our proof with an axiom. Remember, this is just a statement or proposition which is accepted as being true. We often use these as a starting point to a proof. The axiom we’re going to use is that the sum of the interior angles in a triangle is 180 degrees. Remember, the interior angle is the angle made by joining two sides of a polygon that sits inside the shape, as shown here. If we let each of our interior angles be 𝑥, 𝑦, and 𝑧 degrees, we can then say that 𝑥 plus 𝑦 plus 𝑧 equals 180 degrees.

Now, of course, we’re interested in the angle sum of a quadrilateral. That’s a polygon with four sides. So we’re going to add a single straight line from one vertex to another to split our shape into two triangles. Can you see where this line could be? Well, to achieve this, we simply join any vertex to the vertex directly opposite, like this. Notice we’ve now split our quadrilateral into two triangles. We know that the interior angles in this triangle sum to 180 degrees. And the interior angles in this triangle sum to 180 degrees. Then, we see that the interior angles in our quadrilateral are made up of the combined angles in both triangles. That means the interior angles in our quadrilateral sum to 180 plus 180, which equals 360.

And there we have it, a new fact. The sum of the interior angles in a quadrilateral of four-sided polygon is 360 degrees. Note though whilst we used a convex quadrilateral, that is, a four-sided polygon that has interior angles that measure less than 180 degrees each, it also stands for concave quadrilaterals, such as arrowheads. And in fact, had we split our shape up differently, we would have achieved the same result. This geometric proof is actually hugely powerful for deriving a formula to help us calculate the sum of the interior angles in any polygon. Whilst it’s outside the scope of this video to look into this, you might wish to play around with splitting polygons with a variety of number of sides and see what happens. Let’s now have a look at an application of the formula.

The sum of the measures of the angles of a trapezoid is 360 degrees. Write an equation to find the missing measure 𝑥, and then solve it.

We’re given information about the sum of the interior angles. But of course, a trapezoid is a four-sided polygon; it’s a quadrilateral. So we actually know that the interior angles sum to make 360 degrees. Let’s use this information to write an equation. We’re going to begin by adding together the measure of each of our angles. That’s 𝑥 plus 102 plus 116 plus 78. We, of course, know that the sum total of these four angles has to be 360 degrees. So our equation is 𝑥 plus 102 plus 116 plus 78 equals 360.

We need to solve this to find the value of 𝑥. So let’s begin by simplifying the left-hand side. 102 plus 116 plus 78 is 296. So our equation becomes 𝑥 plus 296 equals 360. To solve, we perform the inverse operations. Well, at the moment, we’re adding 296. So the inverse, the opposite to that, is to subtract 296. 𝑥 plus 296 minus 296 is just 𝑥. And 360 minus 296 is 64. So the missing measure 𝑥, our fourth angle at 𝐷, is 64 or 64 degrees.

Now it’s useful to know that with angle problems, there can be more than one way to solve them. In fact, we recall what we know about trapezoids. They have a pair of parallel sides. Our parallel sides have a transversal; that’s 𝐶𝐷. We know that angles 𝑥 and 116 are cointerior or supplementary. That means they add to 180 degrees. So the equation we could have formed would have been 𝑥 plus 116 equals 180. We solve for 𝑥 by subtracting 116. And once again, we find 𝑥 is equal to 64.

Let’s consider another example.

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.

Let’s now consider another example.

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.

In our next example, we’ll look at how to combine the information about the interior angles of a quadrilateral with other angle facts.

From the figure, determine the measure of angle 𝐵𝐸𝐷.

Angle 𝐵𝐸𝐷 is this one. Let’s define it to be equal to 𝑥 degrees. Then, we recall what we actually know about angles in quadrilaterals. Remember, that’s a four-sided polygon. We know that the interior angles in a quadrilateral sum to make 360 degrees. Well, we have a quadrilateral 𝐵𝐶𝐷𝐸. And we know one of its angles is 148 degrees and another is 90. The problem is we can’t find the measure of 𝐵𝐸𝐷 without knowing a third angle. That’s this one. Let’s call that 𝑦 degrees.

We can work out the measure of angle 𝑦 or angle 𝐶𝐵𝐸. We know that angles on a straight line sum to 180 degrees. We see that angle 𝐴𝐵𝐸 is 100 degrees. So 100 plus 𝑦 must be equal to 180. Let’s solve this equation for 𝑦 by subtracting 100 from both sides. And we see that 𝑦 is 180 minus 100, which is 80. So we now know a third interior angle in our quadrilateral. Let’s use this information to form an equation.

The interior angle sum of our quadrilateral is 80 plus 90 plus 148 plus 𝑥, which is equal to 360. 80 plus 90 plus 148 is 318. So our equation is 318 plus 𝑥 equals 360. We’ll solve this equation for 𝑥 by subtracting 318 from both sides. 360 minus 318 is 42. So we find 𝑥 is equal to 42. We therefore say that the measure of angle 𝐵𝐸𝐷 is 42 degrees.

In our very final example, we’ll look at how we can use ratio to find missing angles in a quadrilateral.

The internal angles of a quadrilateral are in the ratio three to four to five to six. What is the largest angle?

We’ve been given some information about the breakdown of four angles in our quadrilateral. So let’s recall what we actually know about the interior or internal angles of a quadrilateral. We know that they sum to 360 degrees. So what we’re going to need to do is share 360 degrees into a given ratio. And there are two ways we can do this. We’ll consider both methods. Let’s consider the first method.

In our first method, we begin by adding the ratios together. That’s three plus four plus five plus six, which is equal to 18. Our second step is to divide our amount, whatever that is, by this number. Here, that’s to divide 360 by 18. In doing so, we’re finding the size of one part in our ratio. 36 divided by 18 is two, so 360 divided by 18 is 20. And this means that one part in our ratio is worth 20 degrees.

The third and final step is to multiply each part of our ratio by this number. But we’re not going to perform that fully. We’re just looking to find the size of the largest angle. In our ratio, that’s represented by the six. Since we know that one part is worth 20 degrees, we’re going to multiply six by 20. Six times 20 is 120. And so the largest angle in the quadrilateral is 120 degrees. Now, what we could do is multiply the other three parts and check these add to make 360. It’s a good way to check our answer.

But let’s consider the other method. The first step in this other method is the same. We add the ratios. The next step is to form a fraction for the parts we are interested in. That might be all the parts, but actually we’re just interested in the largest angle. So that’s the six. Since there are a total of 18 parts, and we’re interested in six of them, the fraction we’re interested in is six eighteenths. The final step here is to find this fraction of the amount.

Well, we know we’re sharing 360 degrees, so we’re going to find six eighteenths of 360. But six eighteenths simplifies to one-third. So we’re going to find one-third of 360. And to find a third, we divide through by three to get 120. And so, given the information about the internal angles of our quadrilateral, we once again see that the largest angle is 120 degrees.

In this video, we’ve learned that we can split a quadrilateral into two triangles by adding a single straight line from one vertex to one opposite it to find the sum of its interior angles. When we do, we find that the result is that the interior angles of a quadrilateral sum to 360 degrees. We can use this fact to solve problems involving missing angles in quadrilaterals by forming and solving equations.

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