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Triangles are polygons with three sides and three angles. Its angles help to classify the triangle, and the relationships between those angles are significant. This lesson explores such angle relationships as well as how they relate to an angle formed outside of the triangle.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

## Investigating the Sum of the Angle Measures

Consider the following triangle. Click and drag vertex to change the shape of the triangle. Pay attention to the angle measures.
What can be said about the sum of the angle measures?
Discussion

## Interior Angles of a Triangle

The interior angles of a triangle are the angles on the inside of the triangle.

Discussion

## Relationship Between the Interior Angles of a Triangle

Triangles can be classified into different types depending on the side lengths or angle measures. Despite all this variety in triangles, the relationship between the interior angles of any triangle can be explained with a single equation.

Rule

## Interior Angles Theorem

The sum of the measures of the interior angles of a triangle is
Based on this diagram, the following relation holds true.

Example

## Bermuda Triangle

For summer vacation, Ali and his parents went on a cruise in the Caribbean Sea. The ship sailed from Miami and headed for San Juan, Puerto Rico. Ali was a little scared because the route went through one side of the Bermuda Triangle 🛳️.

What is the measure of the angle formed at San Juan?

### Hint

What is the sum of the measures of the interior angles of a triangle?

### Solution

Notice that Miami, San Juan, and Bermuda form a triangle for which the measures of two interior angles are given.

Recall that the sum of the measures of the interior angles of a triangle is equal to Use this information to write an equation for the triangle.
Substitute the known measures and solve the equation for
The angle formed at San Juan has a measure of
Example

## A Starry Night

It got dark before the cruise ship arrived in San Juan. Ali noticed that the sky looked much more starry than it does in his hometown. His father explained that it was due to the low light pollution. Ali recognized a triangle made up of three glowing stars.

Write the measure of three interior angles of the triangle.

### Hint

Use the Interior Angles Theorem to set and solve an equation for Then, use the value of to determine each angle measure.

### Solution

The measures of the three interior angles of the triangle depend on The first step is finding the value of The sum of the measures of the interior angles of a triangle is according to the Interior Angles Theorem. Write an equation using this fact.
Solve the equation for
The measures of the angles of the triangle can be found by substituting into each of the expressions.
Angle Measure Substitute Simplify
Discussion

## Angles Outside a Triangle

When the sides of a triangle are extended beyond each vertex, a few more angles are formed. Some of these angles are called exterior angles.

Concept

## Exterior Angles of a Triangle

An exterior angle of a triangle is the angle formed between one side of the triangle and the extension of an adjacent side. Any triangle has six exterior angles, two at each vertex and all formed outside the triangle.
Notice that each exterior angle forms a linear pair with its corresponding interior angle. This means that each exterior angle is supplementary to its interior angle.
Interior Angle Corresponding Exterior Angles Sum of Measures
and
and
and
Discussion

## Remote Interior Angles

In a triangle, a remote interior angle is an interior angle that is not adjacent to a specific exterior angle. Every exterior angle has two remote interior angles. In the diagram, click on each exterior angle to highlight its two corresponding remote interior angles.
The remote interior angles do not share the vertex with their corresponding exterior angle.
Discussion

## Relationship Between an Exterior Angle and Its Remote Interior Angles

In the same way the three interior angles of a triangle are related, the measure of an exterior angle and its two remote interior angles are related.

Rule

## Triangle Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles, or remote interior angles.
Based on the diagram above, the following relation holds true.

Example

## Beach's Day

The day after arriving in Puerto Rico, Ali and his parents took a tour of three of the beautiful Leeward Islands. The captain shared the route plan.

The ship returned to Puerto Rico after visiting Brades.
a Determine the value of which corresponds to the angle that the ship turned to go to Saint Martin island.
b Determine the value of which is the angle at Brades island.

### Hint

a The angle whose measure is is an exterior angle to the triangle formed by the first three islands of the route. Also, the value of is equal to the sum of the measures of the two remote interior angles.
b The angle at Saint Martin island whose measure is is an exterior angle to the triangle formed by the last three islands in the ship's route. The angles whose measures are and are remote interior angles corresponding to the angle.

### Solution

a Start by naming the islands so that it is easier to refer to them. Focus on and ignore the map to have a clearer view.

Notice that is an extension of which means that is an exterior angle to Also, and are remote interior angles corresponding to

The measure of is equal to the sum of the measures of the two remote interior angles. This is because the Triangle Exterior Angle Theorem. Write an equation using this information.
Finally, substitute the corresponding measures to find the value of
b Focus on this time.

In Part A, the measure of is The angle is an exterior angle to Additionally, and are remote interior angles corresponding to the exterior angle.

Once again, use the fact that the measure of an exterior angle is equal to the sum of the measures of the two remote interior angles to write an equation.
Finally, substitute the measures of the angles and solve the equation for
Example

## Angles in the Ship

It is the last day of the cruise and they are on their final island. Ali climbed up a small cliff to take in the beautiful view of the ocean. From the top of the cliff, he could see the docked ship. It is huge! He is curious about some of its measurements.

Assume that the angle that the bow of the ship makes with the water measures What is the value of

### Hint

The angle that is outside the boat is an exterior angle. The angles inside the triangle are remote interior angles corresponding to the exterior angle.

### Solution

Start by labeling the vertices of the triangle and the left endpoint of the extended side. Ignore the photo of the ship to have a clearer view of the diagram.

Notice that is an exterior angle to Also, and are remote interior angles corresponding to Recall that the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. An equation can be written using this information.
The measure of the exterior angle is and the measures of the two remote interior angles were given in terms of Substitute the measure and expressions into the equation and solve it for
Pop Quiz

## Finding the Missing Angle Measure

For each given triangle, find the missing angle measure.

Closure

## Relationship Between the Exterior Angles of a Triangle

Consider a triangle No matter the characteristics of this triangle, the sum of the measures of its interior angles is This fact leads to the question of whether there is a similar relationship between the exterior angles of a triangle.

Recall that the Triangle Exterior Angle Theorem states that the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. As a result, three equations can be written.
Next, add these three equations. The left-hand side of the resulting equation is the sum of the measures of the exterior angles. The right-hand side can be simplified using the fact that the measures of the interior angles add up to
As shown, the sum of the measures of the exterior angles of a triangle, one at each vertex, is equal to Keep in mind that this result holds true for any triangle.