Uncle's Letter

and some lessons.

Hello There!
I see you've been to Billy's world. I am truly grateful to you for helping him escape that awful prison.

And I see that you encountered triangles and found interesting relations between them! Well, I am here to tell you what those actually mean.

Here it goes:

You found those relations because the figures were similar!

Now, what does similarity mean!?

In wordy words, geometrical figures whose dimensions are proportional to one another are called similar figures.

Intuitively, if you take one figure, and make it big, or shrink it, or rotate it, you make a similar figure. They look like siblings who are absurdly similar looking!

Similar Triangles

When specifically concerned with triangles, there are three ways- or criteria- to show that two triangles are similar. You have already gone through these criteria in Billy's world, but here I'll show you mathematically and visually what they are.

You have:

  1. AA (Angle-Angle)
  2. SAS (Side-Angle-Side)
  3. SSS (Side-Side-Side)

Well, their names can tell you a lot!
The first is concerned with two angle pairs. The second- two sides and one angle. The third- all three sides.

Let's see what each of these mean.

AA Similarity Criterion

If you find that the measures of two of the corresponding angle-pairs to be equal, then you know that those two triangles similar based on the AA criterion!

AA

m∠A=m∠Xm∠B=m∠Y⟹△ABC∼△XYZ  (AA criterion)

And it follows that the measures of the other angle-pair are also equal! Why? Because the sum of all the angles in a triangle is 180°!
Note that all equiangular triangles are similar because of this!

SAS Similarity Criterion

If you find that the ratio of two corresponding side-pairs and the measure of the included angle are equal, then you know that those triangles are similar based on the SAS criterion!

SAS

ABXY=BCYZ this is called being “proportional"m∠B=m∠Y⟹△ABC∼△XYZ  (SAS criterion)

SSS Similarity Criterion

This is the rarest of them all!

If you find that all the corresponding side-pairs are proportional, then you know that the triangles are similar based on the SSS Criterion.

SSS

ABXY=BCYZ=ACXZ⟹△ABC∼△XYZ  (SSS criterion)

Note that all equilateral triangles are similar because of this!

So, there you go! You know what similarity is and how triangle similarity can be identified. But of course, there's more to learn!

Never forget that you can always learn more!

Check these links to learn more about math and the world:

  1. 3Blue1Brown
  2. Better Explained
  3. Art of Problem Solving
  4. Numberphile
  5. Nibir's Study Resources and website.

Cheers and Good Luck!

- Billy's Uncle