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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Knowing just a few things about the interior or exterior angles of triangles is sometimes all you need to put all the pieces together. Find out more in this lesson.

Did you ever work on a jigsaw puzzle, devoting hours and hours to putting it together, only to get almost to the end and find out a piece is missing? Maybe it's a piece you'd been looking for on and off for a while. 'There has to be a light blue sky piece somewhere here...'

When we're working with triangles, sometimes we have missing puzzle pieces. Here's an example:

We have a couple angles here, but what is *X*? How are we supposed to figure it out? Did we drop the answer on the floor? Did the dog eat it?

First, we should define what *X* is. If you're looking for a missing puzzle piece, you need to know what it is you need. *X* is an interior angle. An **interior angle** is an angle inside a shape. Since triangles have three angles, they have three interior angles.

In this triangle below, angles *A*, *B* and *C* are all interior angles.

Just as the pieces in a jigsaw puzzle fit together perfectly, the interior angles in a triangle must fit with each other.

The sum of the interior angles is always 180 degrees. In other words, ** a + b + c = 180 degrees**.

Let's prove this. Below are two parallel lines. Let's add a triangle between them. At the top of our triangle, we have three angles based around our line. Let's label them *X*, *Y* and *Z*. These three angles form a straight line, so they add up to what? 180 degrees.

Since we have a parallel line at the bottom of our triangle, we have alternate interior angles. So, the inside angle at the bottom is also equal to *X*. *Z* has an alternate interior angle at the bottom. And look what we did. We just proved that the sum of the interior angles of a triangle is 180 degrees.

Okay, so we know that. How can it help us? Remember our puzzle? In a triangle, you can never be stuck with one missing piece.

In the triangle we were just looking at above, what if we know that angle *X* is 35 degrees and angle *Z* is 60 degrees? Oh, man, if only we know angle *Y*, we'd know them all. But we do! It's 180 - 35 - 60, which is 85.

Here's another one:

In this one, we know angle *X* is 53 degrees. Wait, that's only one angle. We're losing pieces of this puzzle. Is it the cat? Is the cat stealing pieces? No. Look. See this symbol at the bottom left? This means that this angle is 90 degrees. So, we know we have a 53 degree angle and a 90 degree angle. If we subtract 53 and 90 from 180, we get 37 degrees. So, angle *Y* must be 37 degrees. And we can leave the cat alone.

So far, we've been working with fairly straightforward puzzles. But what about those 1,000-piece monsters? Well, okay, maybe that's what you'd call it if you were working with a dodecagon or something, but we're still just talking about triangles. But let's take it outside the box - or outside the triangle. That's right - exterior angles.

An **exterior angle** is an angle created by the side of a shape and a line extended from an adjacent side. In our sample below, *D* is an exterior angle. Now, what do we know about *D*? We know *C* plus *D* is 180. We also know that *A* + *B* + *C* is 180.

That means that ** a + b = d**. Let's state this for the record. An exterior angle is equal to the sum of the non-adjacent interior angles.

In our triangle, we can use *A* and *B* to figure out *D*. If we know *A* is 65 and *B* is 70, we can just add them to get 135, which must be the measure of angle *D*.

Let's try a practice problem. In the triangle below, we know angle *A* is 40 degrees. Whoa. Is this like doing a puzzle while blindfolded? No, wait. See those hash marks? Those indicate that this is an isosceles triangle. Two sides and the angles opposite them are equal. So, *B* and *C* are equal. That means 40 + *B* + *C* = 180. So, *B* + *C* = 140. And if *B* and *C* are equal, then they're each 70. Since *A* + *B* = *D*, we know that *D* is 40 + 70, or 110. We also could've seen that *C* + *D* is 180 and, since we figured out *C*, we could've gotten *D* that way, too.

In summary, we learned that an interior angle is an angle inside a shape, while an exterior angle is an angle made by the side of a shape and a line drawn out from an adjacent side.

The sum of the interior angles of a triangle is always 180. So, if we know two angles, we can always find the third.

An exterior angle is equal to the sum of the non-adjacent interior angles.

Oh, and I never did find that piece of blue sky...

After this lesson, you should have the ability to:

- Define interior and exterior angles
- Explain how to find missing angles using the properties of triangles and angles

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Geometry: High School15 chapters | 160 lessons

- Area of Triangles and Rectangles 5:43
- Perimeter of Triangles and Rectangles 8:54
- How to Identify Similar Triangles 7:23
- Angles and Triangles: Practice Problems 7:43
- Triangles: Definition and Properties 4:30
- Classifying Triangles by Angles and Sides 5:44
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Constructing Triangles: Types of Geometric Construction 5:59
- Properties of Concurrent Lines in a Triangle 6:17
- Go to High School Geometry: Properties of Triangles

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