In Triangle World, a family of triangles just means one triangle in the family is just a bigger or smaller version of another triangle in the family. 

This is kind of the case with humans, where babies and young children are smaller versions of their parents. But unlike humans, each triangle is literally just a scaled up or scaled down version of another triangle in the family. Let me tell you that I think it would be weird for a child of Arnold Schwarzenegger to be just a smaller version of Arnold Schwarzenegger.

Literal Definition

So, how can you find out if two triangles are similar in the triangle world? Well, there are many, many methods, one of which is to use the literal definition. To use this method, we have to have access to all three of the side lengths for both triangles.

Consider the following two triangles:

Now, if we recall our naming convention, these two triangles are called the 3-4-5 and 6-8-10 triangles. But one is just a scaled up version of the other, namely, the 6-8-10. How do we quantify this?

A simple method is to divide the side lengths in each component. That is to say divide the 6, 8, and 10 by the 3, 4, and 5, to get 2, 2, and 2, respectively. To test for similarity, we can just “divide the names by each other” and see if they all agree. This is how we can use the literal definition of similarity to test for it.

Oftentimes in math, however, we have to be smarter than just using the literal definition. Just like in real life, if someone takes every joke literally, he or she is probably not a very fun person. 

An Indirect Method, Pt. 1

So, instead of using the literal definition, how about we forget the side lengths of the triangles above? 

Well, we might be able to squint at it very hard with our eyes or use our rulers, but this method is not going to get us very far. Now that we’ve forgotten the labels, what are we going to do?

Sadly, without any numbers or other markings, we don’t get far mathematically. Let’s re-do this after an explanation of triangle angles.

Interior Angles and Notation

The inside of a triangle has three angles. But what can we say about these angles mathematically?

There is only one thing that is certain for all triangles: these three angles sum up to 180 degrees (the straight line angle).

Okay, but why is this true? Because a smart guy said so? Well, actually, yes. There was a Greek mathematician named Euclid from 300 B.C.E. who basically came up with the rules of (flat) geometry.

And the reason a triangle’s angles sum up to 180 degrees is due to his Fifth Postulate of flatness!

So, yes, if you think long and hard enough, you can make up your own mathematics and theorems!

Anyway, the angles in any triangle add up to 180 degrees, but what more can we say besides that? Well, the second most important angle for triangles is 90 degrees (aka, a right angle), which is the angle of a complete left or right turn.

To convince yourself of this fact, get a sheet of paper and cut out three of the same triangle (any triangle works, but try a triangle with three different angles, and have the edges be straight).

1. Label the three angles A, B, and C on the front face. 

2. Now, label the angles A, B, and C on the back face, while making sure they match with the angles on the front face. 

3. Now, we will fold our back face towards our front face, so that the three tips touch. Word of advice: fold the angle opposite the largest angle down to the side, for easiest folding (the top line has to be parallel to the bottom line). 

4. After folding the three angles such that they meet at the same point, we see that A, B, and C sum up to the straight line angle: 180 degrees. 

Again, this will work for any triangle, but some may be harder to fold than others.

The other types of angles in a triangle are called acute angles or oblique angles. An acute angle is any angle less than 90 degrees (but greater than 0), and an obtuse angle is any angle more than 90 degrees.

There are three classifications of triangles we can have based on their relationship to 90 degrees:

How, why these side lengths imply these are acute, right, or obtuse triangles is an advanced theorem, but we can see the acute, right, and obtuse angles without that theorem.

Homework Questions:

• How many acute angles does an acute triangle have?
• How about a right triangle and an obtuse triangle?
• Why must a right triangle and an obtuse triangle have as many acute angles as they do? (3, obvious; 2 for right and obtuse; because the angles have to add up to 180 degrees, and no angle can be 0 degrees; for example, two right angles can’t happen, or else, we would have a 90 degree, 90 degree, 0 degree, which is not allowed. For an obtuse angle is greater than 90 degrees, for example, 91 degrees, and see that the other two angles have to add up to 89 degrees, since 180 - 91 = 89. But since the other two angles are less than 89 here (or else we would have a 0 degree, 89 degrees, which is not allowed), these two angles are acute.
• What kind of triangle is an equilateral triangle, i.e., one that has three sides of equal length? (Acute, since the angles are equal (since you can rotate as you please), and 180/3 = 60, since there are three angles to split 180 evenly)