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## High school geometry

### Course: High school geometry > Unit 5

Lesson 4: Ratios in right triangles# 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.

## Want to join the conversation?

- Hi, I don't understand why Sal uses opposite leg length/hypotenuse length. Shouldn't it be adjacent length/ hypotenuse length? Can you help me, thx.(9 votes)
- He could have used it but you see we only have the value of opposite side and the hypotenuse. So in order to figure out the ratio he went on with opposite over hypotenuse(10 votes)

- what is a 'leg'? at2:08? THANKS(1 vote)
- A leg of a right triangle is a side other than the hypotenuse. The triangle has one hypotenuse and two legs.

We also use the term for isosceles triangles; a leg is one of the pair of congruent sides.(5 votes)

- How do you know the difference between the adjacent and the opposite leg length?

I always get confused with these two names.

Thanks in advance!(2 votes)- It appears that you can pick out the hypotenuse (longest side opposite the 90 degree angle). Adjacent means next to (like two rooms with a door between them, you walk right in). This forms the reference angle with the hypotenuse. Opposite would be like having to walk across a hall before going in the room, so if you start at the reference angle, you have to walk through the middle of the triangle to find the opposite side.(3 votes)

- what is a hypotenuse or a adjacent length? thanks.(1 vote)
- Hypotenuse is always the longest side of a right triangle across from the 90 degree angle. One of the two acute angles will be a reference angle, so adjacent side forms the reference angle with the hypotenuse. The opposite side requires you to go through the middle of the triangle from the reference angle.(3 votes)

- What is the significance of the 'm' in m∠D?(1 vote)
- It's short for 'measure'. ∠D is the actual angle D, the geometric plane figure. m∠D is the measure of the angle, which is a number like 25º.(3 votes)

- ion even know what happening(1 vote)
- why does it say that these are the approximate ratios for angle measures 25 degrees, 35 degrees, and 45 degrees when it doesn't even add up to 180 degrees?(1 vote)
- That is the angle value for the sin , cos , and tan of the angle(0 votes)

- when I divided 8 by 3.4 it give me 2.35.

How did you find 0.42(0 votes)- We are finding opposite side/hypotenuse, hence we divide 3.4/8 = 0.42.(1 vote)

- ⡍⢎⠵⣉⠶⣉⢆⠣⢆⡱⡘⢦⠱⣘⠦⡑⢎⠴⡁⢎⠴⡡⢆⡳⣜⡸⣴⣯⠶⠿⡼⠧⣿⣼⣧⣽⡮⢧⣷⣜⣦⠵⣨⢒⡔⢦⡘⠴⣈⠆⣍⢢⠱⣌⠲⣌⢢⠱⡌⢆⡱⠌⢦⡑⢎

⡜⢣⢎⡱⢊⠴⡊⡕⢪⠔⡩⢆⠳⣌⠲⣉⢎⢲⡉⣞⣲⣙⣾⡵⠟⠋⢁⠀⢀⠀⠀⢀⢠⡤⠀⠀⠀⠀⠀⠈⠉⠙⠳⠯⣶⣦⡙⣆⠣⢞⡠⣃⠳⣄⠳⡌⢆⠳⡘⠦⡑⡍⢦⡙⢢

⡜⢣⡜⠴⡩⢖⡱⢌⡱⢊⠵⡉⠖⣌⠳⣌⢎⣧⡿⠞⠉⠁⠀⠀⣀⡴⠋⠄⢂⣌⣠⣧⣏⠀⡀⢄⣀⣁⠀⠸⡖⠀⢂⠀⠀⠋⡽⣾⣩⢖⡱⢌⡓⡌⡱⡘⢎⡡⡙⢦⠱⡘⢦⡙⢢

⠜⢣⠘⡥⡑⢎⡰⢃⠼⡡⢎⡹⡘⡔⢫⣼⣟⡏⢀⠀⠄⠈⠀⣴⢏⣤⠶⠚⠉⠁⠀⠀⠈⠗⠋⠉⠀⠈⠙⢿⠟⠓⠲⣬⡁⠀⢰⠉⠻⢾⣵⡮⣔⢣⠱⣉⠦⡑⡍⢆⢣⡙⠆⡍⢆

⢌⠣⡍⢆⡙⢦⡘⡌⢦⡑⢎⠴⡱⣩⢗⣾⡟⠀⠤⠈⠀⣴⠞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⢧⣀⠂⠀⠹⢯⣿⣧⣇⢎⡱⢜⡘⢢⡑⡎⡜⢢

⢌⠳⡘⠦⣉⠦⢱⡘⠦⡜⣌⠲⢥⣛⡾⡿⠃⠈⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠳⣆⠁⡈⢯⣥⡿⣎⠱⣊⠜⣡⠚⠴⡉⢆

⢌⢣⡑⠣⠜⠬⡡⢎⡱⣘⠤⣛⠼⣭⣿⠃⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⡄⢿⡻⣷⣎⠳⣌⠚⡤⢋⠥⡙⢢

⠌⢦⡉⠧⡙⢆⡱⢊⡕⢢⡙⣬⢻⣽⡇⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡇⢨⣿⡾⡷⠱⣌⠓⣌⠣⢎⡱⢡

⢛⠤⡸⢠⡛⢤⢃⢇⡸⢃⠼⣠⣟⣿⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⢻⣿⢣⢇⡘⢧⡘⢧⠘⡄⣃

⠣⡜⢡⠣⡜⢢⠍⢦⠱⣉⠖⣣⣟⣿⢷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣇⢻⣯⣓⠎⡜⢢⡑⢎⡱⡘⠤

⠣⢜⡡⢓⣌⢣⠚⣆⠳⢌⡚⡵⣯⣿⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣣⠏⡜⣡⢚⢢⠱⡘⢥

⢃⠎⡔⢣⠜⣂⠳⡌⢓⡌⢼⡹⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣟⢦⡋⢴⠡⠎⢆⡓⡩⢆

⠌⣎⠱⢌⢎⡔⢣⠜⡢⢍⠲⣝⡿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⡠⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡝⢦⠍⢦⣉⠳⡌⡱⢡⢊

⠜⣠⢋⠜⣢⢌⠣⢎⡱⢎⣽⡾⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⣡⠔⠊⠁⠀⠀⠀⠀⠙⢦⡀⢀⡤⠒⠊⠉⠉⠑⠢⣄⠀⠛⠎⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠺⡅⢮⡑⣌⢓⡰⢃⢇⡊

⡘⡔⣩⢊⡔⢪⡑⢎⠶⣩⢾⡇⢠⠤⡌⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣎⠳⡘⢆⡱⢌⠦⢱⡉⢦⡘

⠰⡑⠦⡱⢌⠣⡜⡌⣞⡱⣿⣿⠈⠀⢻⣣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣠⣧⣄⠀⡾⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣸⠿⣿⡼⣌⢣⡙⢌⡲⡉⢖⡡⠞⠤⣃

⠥⡙⢦⠱⣉⠖⡱⡘⡴⢫⣷⣿⣧⠀⠀⡟⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣇⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣼⡗⠁⣿⡱⢎⡅⡚⢤⢣⢙⢢⡱⢩⠒⡥

⠒⡍⠦⡙⢤⢋⠴⣡⢣⡓⡾⡽⣿⣇⡀⠗⠈⣧⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢠⢾⡎⠀⣸⣿⠱⢣⡘⡱⢊⠦⣉⠦⡑⢣⡙⡰

⠱⡘⢆⡙⢆⡩⢒⠥⢢⠝⣸⢳⣻⢿⣿⣿⣶⣿⣧⠀⠀⠙⠦⣀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡰⠃⣼⠀⣠⣿⢯⡙⢦⠱⣡⢋⠴⢡⠲⣉⠦⡱⢡

⢣⡙⢆⡙⢦⠱⣉⢎⡱⢊⡕⢎⡵⢯⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⡤⠤⠖⠋⠀⢀⣞⠀⠀⢇⣠⣿⣟⢮⡑⢎⡱⢂⠭⣊⠥⢣⡑⢎⡱⢡

⠆⡍⢎⡜⢢⠓⡌⢦⡙⢆⡚⢬⣘⢻⣞⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢰⢚⠒⠒⢲⠒⡦⢤⢤⣤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡽⢦⡙⢢⡑⣍⠲⡡⢎⡱⢌⠎⡔⢣

⢊⡜⢢⠜⣡⢋⡜⢢⡕⢪⠜⡲⢌⣳⢾⣿⣿⣿⣿⣿⣿⣽⣿⣦⣀⠀⠀⠈⠘⢷⣯⣭⣥⣷⣴⠞⠁⠀⠀⠀⢀⣤⣾⣿⣷⣻⣿⣿⣿⣿⣿⡽⢦⠑⣣⠘⡤⢣⡑⢎⡔⣉⠚⡜⠤

⠆⡜⣡⠚⡅⢎⡔⢣⠜⣡⢎⡱⢊⡞⣯⣿⣿⣟⣿⣿⣯⣷⣿⣿⣻⣷⣤⡀⠀⠀⠙⠿⠟⠋⠀⠀⠀⢀⣠⣴⣿⣟⡿⣟⣿⣽⣿⣿⣿⣿⣿⡽⢦⡙⢤⢋⡔⢣⠜⢢⢆⡱⢣⠜⡡

⠱⡘⢤⠓⡜⢢⠜⡡⢞⡰⢢⠜⣣⢽⣳⣿⡿⢾⣿⣿⣿⣿⣟⡾⣷⢯⣟⣿⣷⣶⣤⣤⣤⣤⣤⣶⣾⡿⣟⣯⡷⣿⣽⣻⣟⣿⣿⣿⣿⣿⣿⣝⡣⢜⢢⠣⢜⢢⡙⢆⠎⡔⢣⢎⠱

⢣⠙⢢⡍⡜⢡⡎⢱⠊⡔⢣⡞⣵⣿⣿⣿⣿⣶⣿⣿⣿⢳⣯⣿⡟⠛⠛⠋⠁⠈⠙⣿⣯⣿⣽⣯⣷⣿⢻⣽⢻⣷⣯⣷⡟⣿⣿⣿⣿⣿⢻⣿⣷⠊⣥⠓⡌⡆⢱⠊⡜⡌⢣⡜⢢

⣌⠱⠎⡴⢩⢅⠪⡑⢎⡱⢎⠶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣾⣧⣀⣤⡇⠀⢲⣷⣸⣿⢷⣯⣟⣾⣽⣻⡯⢿⣾⣽⣳⣿⣿⡆⠀⠀⠀⠀⢰⣿⡹⣌⠲⡑⣌⠣⢎⢥⡙⠦⣑⠌

⣄⡋⢖⠡⣃⢎⡱⣉⠦⣑⢎⡿⣹⣿⠛⠛⠛⠛⠛⢛⣿⣿⣿⣾⣟⡿⣿⢷⣿⡿⣟⡿⣯⡿⣾⣽⡟⡉⠀⠠⣤⣽⣿⣻⣿⣿⡇⠀⠐⠀⠀⢈⣿⡳⣌⢣⡑⢦⡙⡌⠦⣌⠣⡜⢨

⢢⡉⢦⡙⡰⢊⠴⡡⠞⡤⢋⡾⡽⣿⣷⣶⣶⣶⣶⣾⣿⣻⣾⣿⣞⣿⡽⣯⣷⢿⣻⣽⡷⣿⣽⣾⡀⣷⣌⣲⣾⣿⢷⣻⣿⣿⣷⣶⣶⣶⣶⣾⣟⡳⣌⠣⡜⡄⢳⢈⠳⣄⠳⣘⠢

⢢⢉⠖⠬⡱⣉⠖⣡⠓⣌⠳⡸⣝⣯⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢾⣽⣻⡷⣿⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣻⣽⣾⣿⣿⣿⣿⣿⣿⣿⢭⡓⢬⠱⡘⡜⡌⠣⢎⠴⠱⣌⠱

⠢⣍⠚⡴⠱⡌⢎⠥⣋⠔⣣⠱⣎⢷⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣷⣿⣿⣿⣿⣽⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣟⢮⣑⢣⢣⠑⢦⣉⠳⣌⠪⡕⣌⠲

⠱⣌⠣⡜⢣⠜⡌⠶⡡⠞⡴⣫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣎⣧⢊⡝⣂⠖⡱⢌⡓⡸⢄⢣

⡑⢆⠳⢌⢣⠚⣌⠣⣕⢫⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⡜⡤⢋⠴⠣⢜⠡⢎⠆

⠸⣌⠳⣌⠣⡝⣠⠓⣬⢷⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣽⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⡰⣉⠎⡱⢊⡱⢊⡜ @thetimmahpaster *(copy paste this message and lets take over khan)(0 votes)

## 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.