A Triangle Is 180 Degrees

A triangle's angles add upwards to 180 degrees considering one exterior angle is equal to the sum of the other 2 angles in the triangle. In other words, the other ii angles in the triangle (the ones that add upward to course the exterior angle) must combine with the 3rd bending to brand a 180 angle.

You know how the angles of a triangle always add together up to 180⁰? Why is that? After all, 180⁰ is the angle that stretches from one side of a straight line to another—so it's kind of weird that that's the number of degrees in the angles of a triangle.

What in the world does a triangle accept to do with a single straight line? Equally information technology turns out, quite a lot. And triangles likewise have a lot to do with rectangles, pentagons, hexagons, and the whole family of multi-sided shapes known as polygons.

We'll meet exactly what I mean past this over the next few weeks. Just for today, we're going to commencement by figuring out exactly why it is that the angles of a triangle always add up to 180⁰. Or so you idea … because we're also going to see that sometimes they don't.

Offset, a review: types of triangles

A quick refresher: at that place are three different types of basic triangles. They are equilateral, isosceles, and scalene.

An equilateral triangle has three sides of the same length
An isosceles triangle has two sides of the same length and one side of a different length
A scalene triangle has three sides of all dissimilar lengths

The length of a triangle'south side directly affects its angles.

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Interior and outside angles

Before we get too far into our story virtually triangles and the total number of degrees in their three angles, there'south one little bit of geometric vocabulary that nosotros should talk about. And that is the difference between an interior and an exterior angle.

The easiest way to describe the difference between these ii things is with an example. Since today'due south theme is the triangle, permit'south talk about the interior and outside angles of a triangle. In brusque, theinterior angles are all the angleswithin the bounds of the triangle. In other words, they're the kind of angles we've been talking about all along.

Theexterior anglesof a triangle are all the angles between ane side of the triangle and the line you get by extending a neighboring sideoutside the premises of the triangle. If yous remember virtually it, yous'll come across that when you add any of the interior angles of a triangle to its neighboring exterior angle, you always get 180⁰—a direct line.

Why do triangles have 180 degrees?

Which brings usa to the main question for today: Why is information technology that the interior angles of a triangle always add together upwardly to 180⁰? As it turns out, you tin figure this out by thinking about the interior and exterior angles of a triangle. To see what I hateful, either grab your imagination or a sheet of paper because it's time for a niggling mathematical arts-and-crafts drawing project.

Start by drawing a right triangle with i horizontal leg, one vertical leg, and with the hypotenuse extending from the height left to the bottom right. Now make a copy of this triangle, rotate it around 180⁰, and nestle information technology up hypotenuse-to-hypotenuse with the original (just as we did when figuring out opens in a new windowhow to find the expanse of a triangle). Finally, make yet some other copy of the original triangle and shift it to the correct so that it's sitting correct next to the newly-formed rectangle. With me so far? If and then, your picture should look like this:

What'south the point of this picture? Take a await at the interior angle at the bottom right of the original triangle (the ane labeled "A"). Now take a look at the ii angles that make up the outside bending for that corner of the triangle (the ones labeled "B" and "C"). Every bit we know, if we add up the interior and exterior angles of one corner of a triangle, nosotros e'er get 180⁰. And our little cartoon shows that the exterior angle in question is equal to the sum of the other 2 angles in the triangle. In other words, the other two angles in the triangle (the ones that add up to form the outside angle) must combine with the angle in the lesser correct corner to make a 180⁰ angle.

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For the sake of simplicity, we've made our drawing using a right triangle. But it turns out that you can make an exactly analogous drawing using any triangle you lot fancy, and you'll e'er terminate upward reaching the same conclusion. Attempt making a few drawings starting with dissimilar triangles of your choosing to meet this for yourself. Every bit an example, hither's some other one that I've made:

The inevitable decision of this game is that the interior angles of a triangle must always add together up to 180⁰. Our lovely and elegant little drawing proves that this must be so.

Or does information technology?

Can triangles accept more than 180 degrees?

Might there be some limitation to our drawing that is blinding united states of america to some other more exotic possibility? Here'south something for y'all to think almost or effort. Procure an uninflated airship, lay it on a flat surface, and describe as close to as perfect of a triangle on it as you can. If you take a protractor handy, it'd be bully to measure and add together up the triangle'due south interior angles and bank check that they're pretty shut to 180⁰.

Now accident upwardly the balloon and have a look at your triangle. What happened to it? If you accept that protractor, try once again to sum upwardly its interior angles. What happened to this sum? Do y'all withal get 180⁰? What does this all mean when it comes to the question of whether or not the interior angles of a triangle ever add together up to 180⁰ as we seem to have found?

Thankfully, I have the answer. Head on over to opens in a new windowside by side week'due south article where we started exploring the strange and wonderful globe known every bit non-Euclidean geometry.