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# 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.
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 $\mathrm{\angle }A$.
$\mathrm{△}ABC$$\mathrm{△}ADE$$\mathrm{△}AFG$$\mathrm{△}AHI$
Opposite leg length$6$$9$$12$$15$
Adjacent leg length$8$
$16$
Hypotenuse length$10$$15$
$25$
Angle A$37\mathrm{°}$$37\mathrm{°}$$37\mathrm{°}$$37\mathrm{°}$
Right angle$90\mathrm{°}$$90\mathrm{°}$$90\mathrm{°}$$90\mathrm{°}$
Last angle
$\mathrm{°}$
$\mathrm{°}$
$\mathrm{°}$
$\mathrm{°}$

Now we're ready to start checking that data for patterns.
Ratio table
Complete the ratio table.
Round to the nearest hundredth.
$\mathrm{△}ABC$$\mathrm{△}ADE$$\mathrm{△}AFG$$\mathrm{△}AHI$
$\frac{\text{adjacent leg length}}{\text{hypotenuse length}}$
$\frac{\text{opposite leg length}}{\text{hypotenuse length}}$
$\frac{\text{opposite leg length}}{\text{adjacent leg length}}$

What did you notice?

## Proving that the pattern works for another angle measure

Proof
Complete the proof that $\frac{AC}{BC}=\frac{FD}{ED}$.
StatementReason
1$\mathrm{\angle }A\cong \mathrm{\angle }F$All right angles are congruent.
2$\mathrm{\angle }B\cong \mathrm{\angle }E$Given
3$\mathrm{△}ABC\sim \mathrm{△}$
similarity
4$\frac{AC}{FD}=\frac{BC}{ED}$Lengths of corresponding sides of similar triangles form equal ratios.
5$\frac{AC}{BC}=\frac{FD}{ED}$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 $25\mathrm{°}$, $35\mathrm{°}$, and $45\mathrm{°}$.
Angle$25\mathrm{°}$$35\mathrm{°}$$45\mathrm{°}$
$\frac{\text{adjacent leg length}}{\text{hypotenuse length}}$$0.91$$0.82$$0.71$
$\frac{\text{opposite leg length}}{\text{hypotenuse length}}$$0.42$$0.57$$0.71$
$\frac{\text{opposite leg length}}{\text{adjacent leg length}}$$0.47$$0.7$$1$
Use the table to approximate $m\mathrm{\angle }J$ in the triangle below.

## Want to join the conversation?

• 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)
• 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,
• 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??
• I agree!
• 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.
• 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.
• What does approximate mean
• 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.