What Should The Sides Of A Triangle Add Up To

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The Quick And Dingy

A triangle's angles add together up to 180 degrees because ane exterior angle is equal to the sum of the other two angles in the triangle. In other words, the other two angles in the triangle (the ones that add upwards to form the exterior angle) must combine with the third angle to make a 180 bending.

You know how the angles of a triangle always add together up to 180⁰? Why is that? Later all, 180⁰ is the bending 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 earth does a triangle take to exercise with a single directly line? Every bit it turns out, quite a lot. And triangles as well take a lot to do with rectangles, pentagons, hexagons, and the whole family unit of multi-sided shapes known as polygons.

Nosotros'll run across exactly what I hateful by this over the next few weeks. But for today, we're going to offset past figuring out exactly why it is that the angles of a triangle always add up to 180⁰. Or so you lot thought … considering we're also going to see that sometimes they don't.

Types of triangles

A quick refresher: there are three unlike types of basic triangles. They are equilateral, isosceles, and scalene.

An equilateral triangle has three sides of the same length
An isosceles triangle has ii sides of the same length and 1 side of a different length
A scalene triangle has three sides of all different lengths

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

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Interior and exterior angles

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

The easiest way to describe the departure between these two things is with an case. Since today's theme is the triangle, let's talk nigh the interior and outside angles of a triangle. In short, theinterior angles are all the angleswithin the bounds of the triangle. In other words, they're the kind of angles we've been talking well-nigh all along.

Interior and Exterior Angles

Theexterior anglesof a triangle are all the angles between ane side of the triangle and the line you become by extending a neighboring sideoutside the bounds of the triangle. If you call back about it, you'll see that when you add any of the interior angles of a triangle to its neighboring exterior bending, you always get 180⁰—a straight line.

How many degrees is a triangle?

Which brings us to the main question for today: Why is it that the interior angles of a triangle always add upwardly to 180⁰? As it turns out, you can figure this out by thinking virtually the interior and outside angles of a triangle. To run into what I mean, either grab your imagination or a sheet of paper because information technology's time for a trivial mathematical arts-and-crafts drawing projection.

Kickoff past cartoon a right triangle with one horizontal leg, one vertical leg, and with the hypotenuse extending from the top left to the bottom right. Now make a copy of this triangle, rotate it around 180⁰, and nestle it up hypotenuse-to-hypotenuse with the original (simply as we did when figuring out how to notice the expanse of a triangle). Finally, brand yet another copy of the original triangle and shift it to the right so that information technology's sitting right adjacent to the newly-formed rectangle. With me then far? If so, your motion-picture show should wait similar this:

Right Triangles

What'south the point of this picture? Take a look at the interior angle at the bottom right of the original triangle (the one labeled "A"). Now take a look at the two angles that make up the exterior angle for that corner of the triangle (the ones labeled "B" and "C"). As we know, if nosotros add up the interior and exterior angles of one corner of a triangle, nosotros always go 180⁰. And our little drawing shows that the outside angle in question is equal to the sum of the other 2 angles in the triangle. In other words, the other 2 angles in the triangle (the ones that add up to form the exterior angle) must combine with the angle in the bottom right corner to brand a 180⁰ angle.

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For the sake of simplicity, we've made our drawing using a right triangle. Only information technology turns out that you tin make an exactly analogous drawing using whatever triangle you fancy, and y'all'll always end upwardly reaching the aforementioned conclusion. Endeavor making a few drawings starting with different triangles of your choosing to see this for yourself. As an example, here'southward another one that I've made:

Non-Right Triangles

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

Or does it?

Do all triangles equal 180 degrees? Can triangles have more than?

Might there be some limitation to our cartoon that is blinding u.s. to some other more exotic possibility? Hither's something for you to remember about or try. Procure an uninflated balloon, lay it on a flat surface, and draw as shut to as perfect of a triangle on information technology as you can. If you accept a protractor handy, it'd be great to measure and add up the triangle's interior angles and check that they're pretty close to 180⁰.

Now blow up the airship and accept a look at your triangle. What happened to it? If yous take that protractor, try once again to sum upward its interior angles. What happened to this sum? Do you however go 180⁰? What does this all mean when it comes to the question of whether or non the interior angles of a triangle always add upwardly to 180⁰ as we seem to accept institute?

Thankfully, I have the answer. Head on over to next week's article where nosotros started exploring the strange and wonderful earth known equally non-Euclidean geometry.