# Interior & Exterior Angles of a Triangle

## What are Interior Angles?

A polygon is a closed two-dimensional shape made up of a number of straight line segments (called its *sides*). In a polygon, an **interior angle** is an angle that is inside the polygon. A triangle has three sides and three interior angles. In the triangle below, the three interior angles are labeled {eq}a {/eq}, {eq}b {/eq}, and {eq}c {/eq}.

Other polygons have more than three interior angles. The number of interior angles in a polygon is always equal to its number of sides. For instance, a hexagon like the one below has six sides and six interior angles.

## Missing Pieces

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?

## What are Exterior Angles?

To form an exterior angle, extend one of the sides past the angle. The angle formed from the extended side and the side that was next to (i.e. adjacent to) the original side is called an **exterior angle**. This angle will always be less than {eq}180^\circ {/eq}.

A polygon with more than three sides also has exterior angles. For example, in the figure below, one exterior angle of a hexagon is labeled.

## Interior vs. Exterior Angles

In the triangle below, an interior angle and an exterior angle that are adjacent are labeled {eq}a {/eq} and {eq}b {/eq}, respectively. Notice that these angles together form a straight angle (i.e. a straight line). Two adjacent angles that form a straight angle are called a **linear pair**. Since a straight angle measures {eq}180^\circ {/eq}, the angles in a linear pair always add up to {eq}180^\circ {/eq}. Angles that add up to {eq}180^\circ {/eq} (whether they are linear pairs or not) are called **supplementary angles**, so linear pairs are always supplementary angles.

For example, in the figure below, if {eq}a=58^\circ {/eq}, then {eq}b=180^\circ-58^\circ=122^\circ {/eq}.

In triangles and in polygons in general, there is a rule that pairs of adjacent interior and exterior angles will always be linear pairs and thus will always be supplementary.

## Interior and Exterior Angles of a Triangle

In polygons, the sum of the interior angles depends only on the number of sides (or the number of angles since they are always the same). For triangles, the sum of the interior angles will always be {eq}180^\circ {/eq}.

In the triangle below, we are given two of the three angles. We can find the missing angle by subtracting those two angles from {eq}180^\circ {/eq}. Thus, the missing third side measures {eq}180^\circ - 80^\circ - 70^\circ = 30^\circ {/eq}.

In contrast, the sum of the exterior angles for *all* polygons, no matter the number of sides/angles, will always be {eq}360^\circ {/eq}. An example of this is shown in the hexagon below. Notice that the exterior angles of this hexagon add up to {eq}360^\circ {/eq}:

{eq}\hspace{2em} 110^\circ+50^\circ+80^\circ+20^\circ+60^\circ+40^\circ = 360^\circ {/eq}

(Note that for any interior angle, there are always two ways to form an adjacent exterior angle. The {eq}360^\circ {/eq} rule will work no matter which of these you choose for each interior angle.)

As another example, we can calculate the exterior angles for the triangle above and see that they add up to {eq}360^\circ {/eq}. In particular, we find that {eq}150^\circ+100^\circ+110^\circ = 360^\circ {/eq}.

We have seen {eq}180^\circ {/eq} appear twice already: namely, in a triangle, the interior angles add up to {eq}180^\circ {/eq}, and interior/exterior angle pairs always add up to {eq}180^\circ {/eq}. That means that two interior angles in a triangle will always add up to the exterior angle of the third angle. For instance, in the triangle above, notice that

{eq}\hspace{2em} \text{Interior} + \text{Interior} = \text{Exterior} {/eq}

{eq}\hspace{2em} 80^\circ+70^\circ = 150^\circ {/eq}

{eq}\hspace{2em} 30^\circ+70^\circ = 100^\circ {/eq}

{eq}\hspace{2em} 30^\circ+80^\circ = 110^\circ {/eq}.

This property will hold for *any* triangle.

## Lesson Summary

In a polygon, an **interior angle** is an angle inside the polygon. In triangles, the sum of the interior angles will always be {eq}180^\circ {/eq}.

To form an exterior angle, extend one of the sides past the angle. The angle formed from the extended side and the adjacent side is called an **exterior angle**. The sum of the exterior angles for all polygons, no matter the number of sides/angles, will always be {eq}360^\circ {/eq}.

Two adjacent angles that form a straight angle (i.e. a straight line) are called a **linear pair**.

Angles that add up to {eq}180^\circ {/eq} are called **supplementary angles**, so linear pairs are always supplementary angles.

In a triangle, two interior angles will always add up to the exterior angle of the third angle.

## Interior Angles

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.

## Practice Problems

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.

## Exterior Angles

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*.

## Practice Problems

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.

## Lesson Summary

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...

## Learning Outcomes

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

To unlock this lesson you must be a Study.com Member.

Create your account

## Missing Pieces

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?

## Interior Angles

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.

## Practice Problems

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.

## Exterior Angles

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*.

## Practice Problems

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.

## Lesson Summary

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...

## Learning Outcomes

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

To unlock this lesson you must be a Study.com Member.

Create your account

#### Do exterior angles add up to 360?

Yes, the sum of the exterior angles for all polygons, no matter the number of sides/angles, will always be 360 degrees.

#### What is interior angle with example?

In a polygon, an interior angle is an angle that is inside the polygon. For instance, a triangle has three interior angles and a hexagon has six interior angles.

#### What is an example of an exterior angle?

To form an exterior angle, extend one of the sides past the angle. The angle that is formed from the extended side and the adjacent side is called an exterior angle.

#### How do you find the exterior angle?

Adjacent exterior and interior angles always add up to 180 degrees (i.e. are supplementary angles). That means you can find the exterior angle by subtracting the interior angle from 180 degrees.

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