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## 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

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.

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

**What did you notice?**

## Proving that the pattern works for another angle measure

## 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.

## Want to join the conversation?

- What the heck? I’m really confused.(51 votes)
- 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.(36 votes)

- I think this whole thing is complicated.(26 votes)
- It is at first. It's hopefully going to get easier though (it better, or I'm going to lose my mind)(15 votes)

- 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!(23 votes)
- 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.(6 votes)

- in
**Proving that the pattern works for another angle measure**, can someone reexplain the last row on the table? i dont understand(6 votes)- 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.(7 votes)

- 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.(9 votes)
- What is the difference between "leg" and just the name?(4 votes)
- 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.(10 votes)

- 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!(2 votes)- 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.(11 votes)

- 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??(7 votes)
- I don't understand how they associated Angle measures with the length of the line segment.(3 votes)
- Wait my bad I forgot to look at the table in top.(6 votes)

- 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.(5 votes)
- 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!(1 vote)