Turn The Pencil Around!

[Warning: This post is quite basic, I mean really basic!]

In an earlier post, we saw a proof without words. On this post we'll see another proof without words. I really loved this when I saw it for the first time and I hope you do too.

Today's topic is really simple: the sum of the angles of a triangle.

We all know that the sum of the angles of any triangle is equal to $180^{\circ}$ . We've all learned how to prove that in school.

Some of you probably tore off the corners of a triangle to verify that in some point in your life [if you haven't done this, try it now!].

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But on this post, we're going to do it in a different way, a way that [hopefully] makes us appreciate the beauty in the simplest of things. Let's get on with it!

We're going to start with a question? What does it mean when you rotate some thing by $180^{\circ}$ ? If we had a pencil lying horizontally and if we rotated it by $180^\circ$ , what would happen?

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The picture above demonstrates what a rotation by $180^\circ$ looks like. Keep that in mind.

We're going to start by drawing a triangle and putting a pencil horizontally below it.

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We have to prove that $\angle CBA+ \angle BAC + \angle ACB=180^\circ$ .

We're going to start by rotating the pencil by $\angle CBA$ .

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Then by $\angle BAC$ ...

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And finally, by $\angle ACB$ ...

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Now if you compare how the pencil was before rotation and what it looks like after being rotated by the angles of triangle $ABC$ , you'll see that the pencil has rotated exactly by $180^\circ$ .

So, $\angle CBA+ \angle BAC + \angle ACB=180^\circ$ [Proved without words!].

This just goes to show that you can look at the simplest of things from a different perspective and enjoy the beauty in them.

Now using the same principle, prove to yourself that the sum of the internal angles of a quadrilateral equals $360^\circ$ . Turn the pencil around!

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Turn The Pencil Around!

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