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Using right triangle ratios to approximate angle measure

Because of similarity, all right triangles with a given acute angle measure have equal ratios between their side lengths. So if we know two of the side lengths of a right triangle, we can figure out the angle measures, too! Created by Sal Khan.

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Video transcript

- [Instructor] We're told here are the approximate ratios for angle measures 25 degrees, 35 degrees, and 45 degrees. So what they're saying here is if you were to take the adjacent leg length over the hypotenuse leg length for a 25-degree angle, it would be a ratio of approximately 0.91. For a 35-degree angle it would be a ratio of 0.82, and then they do this for 45 degrees, and they do the different ratios right over here. So we're gonna use the table to approximate the measure of angle D in the triangle below. So pause this video and see if you can figure that out. All right, now let's work through this together. Now what information do they give us about angle D in this triangle? Well, we are given the opposite length right over here. Let me label that, that is the opposite leg length which is 3.4, and we're also given, what is this right over here? Is this adjacent, or is this the hypotenuse? You might be tempted to say, "Well, this is right next to the angle, "or this is one of the lines, "or it's on the ray that helps form the angle, "so maybe it's adjacent." But remember, adjacent is the adjacent side that is not the hypotenuse. And this is clearly the hypotenuse, it is the longest side, it is the side opposite the 90-degree angle. So this right over here is the hypotenuse, hypotenuse. So we're given the opposite leg length, and the hypotenuse length. And so, let's see, which of these ratios deal with the opposite and the hypotenuse? And if we, let's see, this first one is adjacent and hypotenuse. The second one here is hypotenuse, (laughs) sorry, opposite and hypotenuse. So that's exactly what we're talking about. We were talking about the opposite leg length over the hypotenuse, over the hypotenuse length. So in this case, what is going to be our opposite leg length over our hypotenuse leg length? It's going to be 3.4 over eight, 3.4 over eight, which is approximately going to be equal to, let me do this down here, this eight goes into 3.4. Eight doesn't go into three. Eight goes into 34 four times, four times eight is 32, therefore I subtract, and I can scroll down a little bit, I get a two. I can bring down a zero, eight goes into 20 two times, and that's about as much precision as any of these have. And so it looks like for this particular triangle and this angle of the triangle, if I were to take a ratio of the opposite length and the hypotenuse length, opposite over hypotenuse, I get 0.42. So that looks like this situation right over here. So that would imply that this is a 25-degree, 25-degree angle approximately.