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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!

Note by Mursalin Habib
3 years, 3 months ago

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I hope this is not too basic, even for #CoSinesGroup. Mursalin Habib · 3 years, 3 months ago

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@Mursalin Habib This is very well done, if not just adorable. Math is never "too basic" if you present it well and in an interesting way, as you have done here. Good job! Jacob Erickson · 3 years, 3 months ago

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@Jacob Erickson Thank you so much for your kind words of encouragement! It really means a lot. Mursalin Habib · 3 years, 3 months ago

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@Mursalin Habib Elegantly done. Megh Parikh · 2 years, 11 months ago

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@Mursalin Habib Elegant. Sourav Chaudhuri · 3 years, 3 months ago

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That was a great idea! To quote Feynman- "There are infinite ways in the universe. We just have to hunt them." Simran Gupta · 3 years, 3 months ago

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Cool! John Marvin Macaraeg · 3 years, 3 months ago

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Mursalin, I think what really needs to be noted is your explanation (using words). I have heard in my math class (I teach a grade 7 and 8 class, this is my first year) "why do we needs to use words?". I think the key here is the instructions you give on how to perform the proof. This is a great proof and there is plenty for any beginning student or new teacher (thanks I will be using this, giving you the credit of course) to take away from "basic" proofs that students can feel confident about presenting. What I would like to see is if you could decompose your proof to only the most essential parts and still consider it a proof. Well done all around! Peter Michael · 3 years, 3 months ago

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This opened my eyes O.O Interesting. Jung Un Lee · 3 years, 3 months ago

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Very creative approach. Simply amazing! Nicolae Sapoval · 3 years, 3 months ago

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A creative new solution is always a enjoyment. It's good to note in the moves of the pencil is that every angle is counter-clockwise in other words, positive by convention. Greetings from Chile. Cristopher Andrés Gómez Gallegos · 3 years, 3 months ago

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@Cristopher Andrés Gómez Gallegos I'm glad you enjoyed it! Mursalin Habib · 3 years, 3 months ago

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Got something to teach my younger brother thanks for such an interesting post.. Ritesh G · 10 months ago

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@Ritesh G You're welcome! I'm glad you liked it. Mursalin Habib · 10 months ago

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:) Pirah Sikandar · 2 years, 11 months ago

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This is very basic. Even a class 5 student can understand this. Whoever came up with this idea is great :) Satyam Choudhary · 2 years, 11 months ago

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Nice one.... Archiet Dev · 2 years, 11 months ago

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Simply Fantastic proof! Prasad Nikam · 3 years, 1 month ago

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Nice! I liked how this proof helped encourage visualization! :) Sherry Sarkar · 3 years, 3 months ago

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wow!! interesting n so simple. i like it. Nirbhay Nandan · 3 years, 3 months ago

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Brilliant. Soham Dibyachintan · 3 years, 3 months ago

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WOW! Abhimanyu Swami · 3 years, 3 months ago

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Very interesting read! An elegant proof without words, wonderful. Andrew Tiu · 3 years, 3 months ago

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Nice! Jordi Bosch · 3 years, 3 months ago

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সুন্দর :) Tahsin Ahmed · 3 years, 3 months ago

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Very interesting Mursalin. And I think this is not basic at all. It is such a creative way of proving! Muh. Amin Widyatama · 3 years, 3 months ago

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Very well, Mursalin! Good work! Akshat Jain · 3 years, 3 months ago

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