Video: Finding the Measure of the Largest Angle in a Quadrilateral given a Ratio between the Angles' Measures.

The internal angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. What is the largest angle?


Video Transcript

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.

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