Drawing and tearing proof

How do we know that the angles in a triangle add up to 180 degrees? We have to prove it...

You can follow along with Althea and her friends below. Or you could do the tearing proof first. 

• Draw a triangle.

• Cut it out.

• Cut off the corners.

• Push the corners together, so their vertices (points) are all together.

• Do you see how their outside edges make a line?

• Can you see that the 3 angles must add up to the same degree measure as a line?

• (This isn't exactly a proof. But the fact that it will work no matter how you draw your triangle, that would be a proof. How do we show that? We do the drawing and thinking that Althea and her friends do below.)

And here's a third way to think about it! [Write up the going around the edges thing.]

The proof below actually depends on something called Euclid's 5th Postulate. Wildly, there are alternate geometries, called non-Euclidean, in which we say that the 5th postulate is not true. Mom helps the kids think about spherical geometry, which is one of the models of a non-Euclidean geometry. This link takes you to more about the history of this topic. 

From the (first draft of the) book:

Mom says, “... Let’s all draw a triangle carefully. And let’s try to make our triangles all look different.”

Everyone starts drawing, and Mom keeps talking. Ugh! “Of course we need to start with a definition for what a degree is. And that uses a circle. There are 360 degrees in a circle, because the ancient Babylonians liked multiples of 60, and probably because it’s so close to the number of days in a year.”

She looks around and sees that we each drew a triangle. “Before we get to your triangles, a little more angle background. We can think of 360 degrees as being one revolution. If you faced this way at the center of a circle, and turned around, your eyes would look at each point on the circle, until you got back to where you started. What if we turned halfway? If we draw our original direction as a ray, and our final direction as a ray, what would it look like?”

Aiden actually stands up and turns all the way, and then again, but halfway. He shouts out, “It’s a line!” And he draws a dot on his paper. “That’s me. And that’s the way I was looking at first.” He draws an arrow up. “And that’s the way I was looking after I went halfway around.” And he draws an arrow down.

Mom. “Lovely. So that’s 180 degrees. Pointing from one direction on a line to the other direction.”

Kiara says she would draw it differently. Hers points right at first, and the other side is to the left.

Mom. “That’s actually how we’ll draw it most of the time. Did you all do graphing in your algebra class?” Everyone nods. “ Then you know about the x and y axes. We usually start angles pointing along the positive x-axis. It’s just a convention. We could start any direction.”

That’s funny. Aiden looks relieved.

I hope it’s not rude for me to say this. “Aiden, my Mom thinks it’s great when we’re wrong. We get to think things through and figure them out. Unless you’re worried about what we think. But I bet I’ll screw up more than you will. Turns out you were right on that But even if you had been wrong, you did a picture, which is more than I did.”

Aiden looks at me funny. I wonder if I should have kept my mouth shut. “Sorry.”

Aiden. “No, no. Thanks. I get in trouble so much at school, I guess I get worried.”

He pauses, and then gets revved up again. “Wait! If it’s angles from the origin, I know that! I use those when I’m making new games up.”

Mom asks, “What kinds of games?”

Aiden looks a little embarrassed. “So far, mostly it’s people shooting zombies. But I want to do tons more.”

Mom’s eyebrows go up. “Wow. I don’t know how to do that. If you ever want to give me a lesson, I’d be interested.”

Aiden looks surprised. “Cool. I’m down for that, Ms. DeRivera.”

Mommy looks a little startled. “Aiden, are you comfortable calling me Annie, or Ms. Annie? If you are, that’s my preference.”

Aiden says, “Sure, Miss Annie! Oh, is miss ok?”

Mom, with a big smile. “Yep. So Aiden and Kiara have shown me 180 degrees. Althea and Sofia, are you ok with that idea?”

We both nod.

Mom has looked at all our triangles and notices, “I see that we all put one side flat at the bottom. We could make a triangle pointing down like this, but it’s actually convenient that we all have a flat bottom side. Now if you draw a line through the top point or vertex that’s parallel to the bottom side, we can talk about some angles that will be equal. Actually, we’ll want to extend all 3 sides of the triangle, too.”

“We usually use Greek letters for the names of angles. For the angles in a triangle, we typically use α, β, and γ. Those are called alpha, beta, and gamma. But if we were focusing on just one angle, we’d usually use θ. That’s called theta. That would be instead of the x we usually use for our variable.” She stops and writes those on the board.

“So let’s put alpha at the bottom left, beta at the bottom right, and gamma in the top angle of the triangle. Do yours all look like this?”

She looks around and nods. “Good. Now this is the part where I’m not sure if you need more geometry experience. Can you see some other angles in your picture that must be the same size as alpha?”

I’ve gotta stop thinking about Mom and look at this. Hmm. I think the one kind of opposite alpha must be the same size. And opposite beta and gamma too! I wonder if there are any others…

Aiden kind of yelps. “Miss Annie, is this right?! There are so many if it is.”

Mom nods. I try not to look at his drawing, and I try to find more. Hmm, if the lines had to be parallel, that must be important somehow, and it has nothing to do with the ones I found. If I slide alpha up to the other line, then I guess this one would be alpha too!

Mom. “We agreed that a full rotation is 360 degrees, and that halfway around makes a line, and that that would be 180 degrees.”

Aiden is nodding like crazy. “And so these 3 angles add up to 180 degrees!” He’s pointing on top.

I add, “Or the ones on the bottom of that top line. Those add up to 180 degrees too.”

We have another whiteboard on the wall in the sunroom. Mom copies Aiden’s picture, and says “Aiden says these 3 add up to 180, and Althea says these 3 add up to 180. Kiara?”

Kiara had her hand up. “So alpha plus beta plus gamma is 180 degrees. And that’s why the angles in a triangle add up to 180!”

Mom writes while she’s talking. “Yep. But if you don’t like this way of seeing it, there’s another way to see it. Can you all cut out your triangles?” And she hands us all scissors. Wow, Mom must have really prepared for us. I didn’t realize she’d be doing anything extra besides spending some of this time with us. We cut them out pretty quickly.

Mom. “Now rip off each of the corners, and put them next to each other, with all the points kind of toward the bottom.”

Sofia. “Oh! No way! This is way better! The angles totally make a line!” She pauses. “Huh. So there’s a reason for the angles in the triangle adding up to 180. That’s cool.” 

And now you can solve this: