So now, we have to ask ourselves this question: “if similarity is about the sides of triangles, but we don’t know the three sides, what else can we use?” An important answer to this question is “try the simplest thing that can possibly work.” Maybe don’t do this if you’re doing art. 

If we don’t have sides, we at least need some angles, and in this lesson, we’re going to get plenty of them.

An Indirect Method, pt. 2: Equiangularity means similar (AAA)

Woah. What is “equiangularity?” It just means “equal angles.” Basically, if two triangles have the exact same three angles, they are called “equiangular.” I want to show you that two triangles having the exact same angles means that they are similar. In addition, the angles match up opposite to where the sides are proportional.

 An actual proof of this needs about six dense chapter worth of facts about triangles and parallel lines, which would take a little too long, so instead, let’s just use this simple scaling activity:

1. Pick any possible combination of 3 triangle sides (preferably all 5 or under). Recall the inequality to make this happen. (like 1-4-4, for example)
2. Scale these up to the 2 times version (2-8-8, in the same example)
3. Cut out the triangles with those dimensions (fiddle around with this until the lines meet and are the correct lengths) (1” - 4” - 4”, for example)
4. See that the angles are the same.
   

So any two triangles having the same three angles means they are similar, i.e., an identical copy or a scaled up or scaled down version of each other.

Similarity means equianglarity

Let’s get down to business. If I have a similar triangle, then all three of the angles will be the same as the original triangle. In English: if you scale a triangle’s sides up or down by the same amount, you won’t change the angles.

Let’s prove why this is true. To do this, I’m going to copy a smart man named Euclid. From his Elements Proposition 5, we assume we have two similar triangles 

And to note the ratios of similarity, we’ll say AB / DE = BC / EF = CA /FD.

And then we’re going to make a copy of ABCthat has all the same angles, BUT scale it down so that it shares the common side EF with DEF. (We also flip it upside down.) So we’ll call this triangle EFG, and this triangle’s angles will be ∠GEF = ∠ABC, ∠EFG = ∠BCA, ∠FGE = ∠CAB.

But equiangular triangles are similar! So, ABCis similar to EFGby the sides corresponding to opposite angles (this was what we said in the previous section!). But these ratios are the exact same as the ones in DEF, so DEFand GEFare congruent, and because of this, the angles, correctly ordered, are the same.

So, being similar means that two triangles have the same three angles.

Conclusion:

- Equiangularity (having the same three angles) means similarity.

- Similarity (being a copy or a scaled up or scaled down version) means equiangularity.

- These two concepts are the exact same thing, as we logically proved (at least one half), so we can use these terms interchangeably.