What the heck? I’m really confused.

It's basically saying that if you know one angle of a right angle triangle (other than the 90 degree angle) you can use this to deduce what the ratios of the side lengths are, whether that be opposite / hypotenuse (sin), Adjacent / Hypotenuse (cos), or opposite / adjacent (tan). Once you know the ratios, as soon as you have a side length, you can use SOHCAHTOA to find all of the side lengths, which should be covered in other lessons. At the moment all that you really need to know is that all triangles with the same angles will have the same side length ratios.

I think this whole thing is complicated.

It is at first. It's hopefully going to get easier though (it better, or I'm going to lose my mind)

I have been using Khan Academy for years! Returning to Khan Academy after a long break to refresh my memory on trigonometry. This trigonometry course is all over the place. When I first studied trigonometry on this site a few years ago, the flow and structure of the unit was clearer and progressed incrementally. In this unit, there are some simple concepts followed by complex concepts. This page needs to be restructured to make it easier for learners, as it was a few years ago!

I am so happy for you, @Kasim Ahmed! Trigonometry has helped me a lot too{i used to be called a dummy because i was terrible at math, but now i am at the top of my class/ all thanks to Khan-Academy} Have a BLESSED day! M.L.M.

in *Proving that the pattern works for another angle measure*, can someone reexplain the last row on the table? i dont understand

It says that if we multiply both sides of the equation in step 4 (𝐴𝐶∕𝐹𝐷 = 𝐵𝐶∕𝐸𝐷) by some factor 𝑘, we end up with the equation in step 5 (𝐴𝐶∕𝐵𝐶 = 𝐹𝐷∕𝐸𝐷) This gives us the system of equations 𝑘⋅𝐴𝐶∕𝐹𝐷 = 𝐴𝐶∕𝐵𝐶 𝑘⋅𝐵𝐶∕𝐸𝐷 = 𝐹𝐷∕𝐸𝐷 We can solve for 𝑘 in either equation. In both cases we get 𝑘 = 𝐹𝐷∕𝐵𝐶, which is our answer.

What is the difference between "leg" and just the name?

There isn't any difference! Leg is just a name to describe any of the the 2 sides that aren't the hypotenuse which are the opposite side/leg and adjacent side/leg.

I don't understand how they associated Angle measures with the length of the line segment.

Wait my bad I forgot to look at the table in top.

So wait I don't quite get it. At the end we conclude that now we have a way to figure out all the side lengths of a triangle if we know one side length and one of the angle measures. I understand up to that point. But I don't know, say if I know these values in a given triangle, what to do from there. I'm just given these pre-completed tables. So are we going to learn how to get these numbers I see in the table later on? Or did I miss something in these lessons? Thanks!

If you include that fact that it has to be a right triangle, then you are correct that having a side and an angle (or two sides) will allow you to find all other parts of the triangle (approximations for the most part). However, we generally find the numbers on the table with use of a calculator, not by hand. There should be trig functions on any graphing calculator. You will progress to non-right triangles later, but you need more information for them.

Main content

High school geometry

Course: High school geometry > Unit 5

Lesson 4: Ratios in right triangles

Side ratios in right triangles as a function of the angles

Google Classroom

By similarity, side ratios in right triangles are properties of the angles in the triangle.

When we studied congruence, we claimed that knowing two angle measures and the side length between them (Angle-Side-Angle congruence) was enough for being sure that all of the corresponding pairs of sides and angles were congruent.

How can that be? Even with the Pythagorean theorem, we need two side lengths to find the third. In this article, we'll take the first steps towards understanding how the angle measures and side lengths give us information about each other in the special case of right triangles.

This is a great opportunity to work with a friend or two. The goal of this article is to find and discuss patterns, not to spend a bunch of time calculating. Try splitting up the work so there's more time to talk about what you see!

Let's look for patterns

First, we'll collect some data about a set of triangles.

How are the four triangles related?

The triangles are

according to the

criterion.

Measurement table

Here are those triangles again.

Complete the table of measurements relative to $∠ A$ ‍.

[Remind me what sides are opposite, adjacent, and the hypotenuse.]

	$△ A B C$ ‍	$△ A D E$ ‍	$△ A F G$ ‍	$△ A H I$ ‍
Opposite leg length	$6$ ‍	$9$ ‍	$12$ ‍	$15$ ‍
Adjacent leg length	$8$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	$16$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍
Hypotenuse length	$10$ ‍	$15$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	$25$ ‍
Angle A	$37 °$ ‍	$37 °$ ‍	$37 °$ ‍	$37 °$ ‍
Right angle	$90 °$ ‍	$90 °$ ‍	$90 °$ ‍	$90 °$ ‍
Last angle	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $°$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $°$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $°$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍ $°$ ‍

[Show me the measurement table.]

Now we're ready to start checking that data for patterns.

Ratio table

Complete the ratio table.
Round to the nearest hundredth.

	$△ A B C$ ‍	$△ A D E$ ‍	$△ A F G$ ‍	$△ A H I$ ‍
$\frac{adjacent leg length}{hypotenuse length}$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍
$\frac{opposite leg length}{hypotenuse length}$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍
$\frac{opposite leg length}{adjacent leg length}$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍	Your answer should be an integer, like $6$ ‍ a simplified proper fraction, like $3 / 5$ ‍ a simplified improper fraction, like $7 / 4$ ‍ a mixed number, like $1 3 / 4$ ‍ an exact decimal, like $0.75$ ‍ a multiple of pi, like $12 pi$ ‍ or $2 / 3 pi$ ‍

[Show me the ratio table.]

What did you notice?

[Talk to your friend about what you noticed first, then look here.]

Proving that the pattern works for another angle measure

Proof

Complete the proof that $\frac{A C}{B C} = \frac{F D}{E D}$ ‍.

	Statement	Reason
1	$∠ A ≅ ∠ F$ ‍	All right angles are congruent.
2	$∠ B ≅ ∠ E$ ‍	Given
3	$△ A B C \sim △$ ‍	similarity
4	$\frac{A C}{F D} = \frac{B C}{E D}$ ‍	Lengths of corresponding sides of similar triangles form equal ratios.
5	$\frac{A C}{B C} = \frac{F D}{E D}$ ‍	Multiply both sides by .

Conclusion of proof

What did we prove?

What triangles did we prove it for?

What did we conclude?

If two right triangles have an acute angle measure in common, they are similar by angle-angle similarity. The ratios of corresponding side lengths within the triangles will be equal. So the ratio of the side lengths of a right triangle just depends on one acute angle measure.

Why will this be useful?

Before, we could use the Pythagorean theorem to find any missing side length of a right triangle when we knew the other two lengths. Now, we have a way to relate angle measures to the right triangle side lengths. That allows us to find both missing side lengths when we only know one length and an acute angle measure. We can even find the acute angle measures in a right triangle based on any two side lengths.

Extension 1.1

Given the measure of an acute angle in a right triangle, we can tell the ratios of the lengths of the triangle's sides relative to that acute angle.

Here are the approximate ratios for angle measures

25 °

35 °

, and

45 °

Angle	$25 °$ ‍	$35 °$ ‍	$45 °$ ‍
$\frac{adjacent leg length}{hypotenuse length}$ ‍	$0.91$ ‍	$0.82$ ‍	$0.71$ ‍
$\frac{opposite leg length}{hypotenuse length}$ ‍	$0.42$ ‍	$0.57$ ‍	$0.71$ ‍
$\frac{opposite leg length}{adjacent leg length}$ ‍	$0.47$ ‍	$0.7$ ‍	$1$ ‍

Use the table to approximate $m ∠ J$ ‍ in the triangle below.

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Bailey White
Posted 2 years ago. Direct link to Bailey White's post “What the heck? I’m really...”
What the heck? I’m really confused.
Button navigates to signup pageComment on Bailey White's post “What the heck? I’m really...”
(51 votes)
Answer
- PiFace3.1415
  Posted a year ago. Direct link to PiFace3.1415's post “It's basically saying tha...”
  It's basically saying that if you know one angle of a right angle triangle (other than the 90 degree angle) you can use this to deduce what the ratios of the side lengths are, whether that be opposite / hypotenuse (sin), Adjacent / Hypotenuse (cos), or opposite / adjacent (tan). Once you know the ratios, as soon as you have a side length, you can use SOHCAHTOA to find all of the side lengths, which should be covered in other lessons.
  At the moment all that you really need to know is that all triangles with the same angles will have the same side length ratios.
  Comment on PiFace3.1415's post “It's basically saying tha...”
  (36 votes)
mrsdanielgray
Posted 6 months ago. Direct link to mrsdanielgray's post “I think this whole thing ...”
I think this whole thing is complicated.
Button navigates to signup pageComment on mrsdanielgray's post “I think this whole thing ...”
(26 votes)
Answer
- C4LOwenZ
  Posted 5 months ago. Direct link to C4LOwenZ's post “It is at first. It's hope...”
  It is at first. It's hopefully going to get easier though (it better, or I'm going to lose my mind)
  Comment on C4LOwenZ's post “It is at first. It's hope...”
  (15 votes)
Kasim Ahmed
Posted 7 months ago. Direct link to Kasim Ahmed's post “I have been using Khan Ac...”
I have been using Khan Academy for years! Returning to Khan Academy after a long break to refresh my memory on trigonometry. This trigonometry course is all over the place. When I first studied trigonometry on this site a few years ago, the flow and structure of the unit was clearer and progressed incrementally. In this unit, there are some simple concepts followed by complex concepts. This page needs to be restructured to make it easier for learners, as it was a few years ago!
Button navigates to signup pageComment on Kasim Ahmed's post “I have been using Khan Ac...”
(23 votes)
Answer
- Marissa.L.Medina
  Posted 7 months ago. Direct link to Marissa.L.Medina's post “I am so happy for you, @K...”
  I am so happy for you, @Kasim Ahmed!
  Trigonometry has helped me a lot too{i used to be called a dummy because i was terrible at math, but now i am at the top of my class/ all thanks to Khan-Academy}
  Have a BLESSED day!
  M.L.M.
  Comment on Marissa.L.Medina's post “I am so happy for you, @K...”
  (6 votes)
shelley
Posted a year ago. Direct link to shelley's post “in *Proving that the patt...”
in Proving that the pattern works for another angle measure, can someone reexplain the last row on the table? i dont understand
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Jerry Nilsson
  Posted a year ago. Direct link to Jerry Nilsson's post “It says that if we multip...”
  It says that if we multiply both sides of the equation in step 4
  (𝐴𝐶∕𝐹𝐷 = 𝐵𝐶∕𝐸𝐷)
  by some factor 𝑘, we end up with the equation in step 5
  (𝐴𝐶∕𝐵𝐶 = 𝐹𝐷∕𝐸𝐷)
  
  This gives us the system of equations
  𝑘⋅𝐴𝐶∕𝐹𝐷 = 𝐴𝐶∕𝐵𝐶
  𝑘⋅𝐵𝐶∕𝐸𝐷 = 𝐹𝐷∕𝐸𝐷
  
  We can solve for 𝑘 in either equation.
  In both cases we get 𝑘 = 𝐹𝐷∕𝐵𝐶,
  which is our answer.
  Button navigates to signup page
  (7 votes)
Black Cat
Posted 2 years ago. Direct link to Black Cat's post “Why are some of the instr...”
Why are some of the instructions so vague? For example, it gives the approximate length of the side after showing what process to use but does not show how we got to the results.
Button navigates to signup pageButton navigates to signup page
(9 votes)
Answer
San(gharsha)
Posted 3 years ago. Direct link to San(gharsha)'s post “What is the difference be...”
What is the difference between "leg" and just the name?
Button navigates to signup pageComment on San(gharsha)'s post “What is the difference be...”
(4 votes)
Answer
- rrsaroya
  Posted 2 years ago. Direct link to rrsaroya's post “There isn't any differenc...”
  There isn't any difference! Leg is just a name to describe any of the the 2 sides that aren't the hypotenuse which are the opposite side/leg and adjacent side/leg.
  Button navigates to signup page
  (10 votes)
Stephen Earley
Posted 2 years ago. Direct link to Stephen Earley's post “So wait I don't quite get...”
So wait I don't quite get it. At the end we conclude that now we have a way to figure out all the side lengths of a triangle if we know one side length and one of the angle measures. I understand up to that point. But I don't know, say if I know these values in a given triangle, what to do from there. I'm just given these pre-completed tables. So are we going to learn how to get these numbers I see in the table later on? Or did I miss something in these lessons?
Thanks!
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- David Severin
  Posted 2 years ago. Direct link to David Severin's post “If you include that fact ...”
  If you include that fact that it has to be a right triangle, then you are correct that having a side and an angle (or two sides) will allow you to find all other parts of the triangle (approximations for the most part). However, we generally find the numbers on the table with use of a calculator, not by hand. There should be trig functions on any graphing calculator. You will progress to non-right triangles later, but you need more information for them.
  Button navigates to signup page
  (11 votes)
gamag299th
Posted 7 months ago. Direct link to gamag299th's post “What in the congruent rel...”
What in the congruent relationship of pythagoras and a rectangular prism? These instructions don't teach you. Where do I learn what a side-side-side angle and a angle-angle is??
Button navigates to signup pageComment on gamag299th's post “What in the congruent rel...”
(7 votes)
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- Rediet N.
  Posted 5 months ago. Direct link to Rediet N.'s post “I agree!”
  I agree!
  Button navigates to signup page
  (1 vote)
avnish.devekar.438
Posted 5 months ago. Direct link to avnish.devekar.438's post “I don't understand how th...”
I don't understand how they associated Angle measures with the length of the line segment.
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- avnish.devekar.438
  Posted 5 months ago. Direct link to avnish.devekar.438's post “Wait my bad I forgot to l...”
  Wait my bad I forgot to look at the table in top.
  Button navigates to signup page
  (6 votes)
caldwelljt
Posted 8 months ago. Direct link to caldwelljt's post “I can't help but feel thi...”
I can't help but feel this was horribly explained (for how I think)... I know this is just one of those things i'll have to look away and come back to in a couple hours to get the crux of it.
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(5 votes)
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- Diego Bernal
  Posted 7 months ago. Direct link to Diego Bernal's post “This is just an overview....”
  This is just an overview. You will find these concepts explained in more depth in the high school geometry course here on Khan Academy. Hope this helps!
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  (1 vote)