body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

{{ 'ml-label-loading-course' | message }}

{{ toc.name }}

{{ toc.signature }}

{{ tocHeader }} {{ 'ml-btn-view-details' | message }}

{{ tocSubheader }}

{{ 'ml-lesson-btn-finish-worksheet' | message }}

{{ 'ml-toc-proceed-mlc' | message }}

{{ 'ml-toc-proceed-tbs' | message }}

An error ocurred, try again later!

Chapter {{ article.chapter.number }}

{{ article.number }}.

{{ article.displayTitle }}

{{ article.intro.summary }}

Show less Show more

{{ ability.displayTitle }}

Lesson Settings & Tools

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}

{{ item.file.title }}
{{ presentation }}

No file copyrights entries found

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.

Angles
Alternate Interior Angles
Alternate Interior Angles Theorem
Triangles

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.

Triangle with a movable vertex

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

The triangle ABC with movable vertices A, B, and C and the angle measures written

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 🛳️.

Error loading image.

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

SubstituteValues

Substitute values

Add terms

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.

Error loading image.

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

CommutativePropAdd

Commutative Property of Addition

Add and subtract terms

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.

Triangle with exterior angles labeled

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.

A triangle with one exterior angle and its two 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.

Triangle with an exterior angle marked

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.

Error loading image.

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

SubstituteValues

Substitute values

Add terms

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

SubstituteValues

Substitute values

Rearrange equation

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.

Error loading image.

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

SubstituteExpressions

Substitute expressions

Distribute

Add terms

Rearrange equation

Pop Quiz

Finding the Missing Angle Measure

For each given triangle, find the missing angle measure.

Random triangles

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

Add terms

Factor out

Multiply

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.

{{ grade.displayTitle }}

Edit Lesson

Loading content