We won’t be dealing with more complicated triangles on curved surfaces like a donut or a sphere, but there’s plenty to gain from studying flat triangles, too! Triangles come in different shapes and sizes (well, besides the non-triangle ones, anyway), but they’re also alike in many other ways. Let’s take a look at some of those differences and similarities! 

 Every triangle has three sides and three angles (why else would they be called triangles?), but how do we tell any of them apart or say they’re the same?  

Well, let’s ask how you might tell two friends apart. Maybe one is taller? Or one has longer hair? Maybe one of them has bigger muscles?   

While your eyes might find a perfectly acceptable way of telling two people apart, a simple one is to know their names (presumably these are different!)  

Similarly, every triangle has a name! They’re just a little more number-ey and mathematical, and they also follow certain rules. For example, if I want to name a triangle by its three sides, I might give it a name like 3-4-5 or 3-3-3. However, certain other names won’t work, like 1-2-3 or 2-2-5 (we’ll see why in the next lesson). And unlike humans, any two triangles with the same name are actually the same triangle!  

Congruence:  

What does it mean to be the same triangle (or, in fancy math talk, being “congruent”)?  

Well, imagine this scenario: if you were to walk with your friends down the sidewalk and take a right turn, would they be confused as to who you are? Probably not. Clearly, in this scenario, you walked yourself forward and then rotated to the right, but no one would call you by a different name. You’re still the same human!


Similarly, with a triangle, you can move them up, right, down, left, and right, or even rotate them, but even after they move like this, we would still call them the same triangle. So, if there are two triangles, the test to see if they are the same triangle is the perfect overlap test.  

To illustrate this point, let’s make two 6-8-10 triangles. To do this, you will need 2 sheets of paper (8.5” x 11”), a protractor, scissors, a pencil, and a ruler.    

1. Have the two sheets of paper perfectly overlap each other, since we are making two triangles.  
2. Towards the vertical right edge of the top paper, draw a 8” straight line.  
3. At 90 degrees (measure with protractor) from the bottom point of the 8” vertical line just drawn, draw a 6” horizontal line extending left.  
4. If done correctly, the diagonal line connecting the left endpoint of the 6” horizontal line and the top endpoint of the 8” vertical lines should be 10” in length. Draw this diagonal.  
5. Cut the triangle just drawn along the lines.  

Now, you should have two 6-8-10 triangles. Right now, these triangles are perfectly overlapping, and you can check both sides to see that the triangles we just cut aren’t overlapping each other. For added fun, you can color the two sides (ideally, with something that doesn’t smudge/won’t get messy).  

On a large enough table or flat surface, separate the two triangles and rotate them in any manner (even a 3D rotation, by picking a triangle up and placing it down on the other side, if you want to test creativity). See if your student can get them to perfectly overlap again, with at most one triangle picked up as a final step (i.e., they should try to slide and rotate on the flat surface as much as possible).  

Now, cut any triangle that is not the 6-8-10 triangle, and scramble the three triangles. When you line them up, the two congruent triangles will always overlap when you look at both sides, but the non-congruent triangle out will not, no matter how you rotate it or move it.  

Again, to summarize, being a congruent (or “the same”) triangle means the two triangles can pass the perfect overlap test. Since this is an introduction, we are not going to use formal symbols used, like for congruence or triangle notation, since this is about building intuition.