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Pre-Algebra View details

5. Angles in Triangles

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eCourses /

Pre-Algebra /

Chapter 9

Angles in Triangles

This lesson explores the relationships between various angles in a triangle. It covers interior and exterior angles and explains the triangle exterior angle theorem, helping learners understand how these angles interact and how to solve related geometric problems effectively.

Lesson Settings & Tools

	13 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Angles in Triangles

Slide of 13

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

A

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.

An acute triangle ABC with interior angles labeled as angle ABC, angle BCA, and angle CAB.

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

18 0^{\circ} .

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

Based on this diagram, the following relation holds true.

$m ∠ A + m ∠ B + m ∠ C = 18 0^{\circ}$

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

18 0^{\circ} .

Use this information to write an equation for the triangle.

m ∠ M + m ∠ S + m ∠ B = 18 0^{\circ}

Substitute the known measures and solve the equation for

m ∠ S .

m ∠ M + m ∠ S + m ∠ B = 18 0^{\circ}

SubstituteValues

Substitute values

5 6^{\circ} + m ∠ S + 6 1^{\circ} = 18 0^{\circ}

AddTerms

Add terms

m ∠ S + 11 7^{\circ} = 18 0^{\circ}

SubEqn

$LHS - 11 7^{\circ} = RHS - 11 7^{\circ}$

m ∠ S = 6 3^{\circ}

The angle formed at San Juan has a measure of

6 3^{\circ} .

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 $x .$ Then, use the value of $x$ to determine each angle measure.

Solution

The measures of the three interior angles of the triangle depend on

x .

The first step is finding the value of

x .

The sum of the measures of the interior angles of a triangle is

18 0^{\circ},

according to the Interior Angles Theorem. Write an equation using this fact.

3 x + (6 x + 1) + (5 x - 3) = 180

Solve the equation for

x .

3 x + (6 x + 1) + (5 x - 3) = 180

CommutativePropAdd

Commutative Property of Addition

3 x + 6 x + 5 x + 1 - 3 = 180

AddSubTerms

Add and subtract terms

14 x - 2 = 180

AddEqn

$LHS + 2 = RHS + 2$

14 x = 182

DivEqn

$LHS / 14 = RHS / 14$

x = 13

The measures of the angles of the triangle can be found by substituting

x = 13

into each of the expressions.

Angle Measure	Substitute $x = 13$	Simplify
$3 x^{\circ}$	$3 (13)^{\circ}$	$3 9^{\circ}$
$(6 x + 1)^{\circ}$	$(6 (13) + 1)^{\circ}$	$7 9^{\circ}$
$(5 x - 3)^{\circ}$	$(5 (13) - 3)^{\circ}$	$6 2^{\circ}$

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
$∠ 7$	$∠ 1$ and $∠ 2$	$m ∠ 1 + m ∠ 7 = 18 0^{\circ}$ $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$
$∠ 8$	$∠ 3$ and $∠ 4$	$m ∠ 3 + m ∠ 8 = 18 0^{\circ}$ $m ∠ 4 + m ∠ 8 = 18 0^{\circ}$
$∠ 9$	$∠ 5$ and $∠ 6$	$m ∠ 5 + m ∠ 9 = 18 0^{\circ}$ $m ∠ 6 + m ∠ 9 = 18 0^{\circ}$

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.

$m ∠ P C A = m ∠ A + m ∠ B$

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

x,

which corresponds to the angle that the ship turned to go to Saint Martin island.

b Determine the value of

y,

which is the angle at Brades island.

Hint

a The angle whose measure is

x^{\circ}

is an exterior angle to the triangle formed by the first three islands of the route. Also, the value of

x

is equal to the sum of the measures of the two remote interior angles.

b The angle at Saint Martin island whose measure is

7 2^{\circ}

is an exterior angle to the triangle formed by the last three islands in the ship's route. The angles whose measures are

x^{\circ}

and

y^{\circ}

are remote interior angles corresponding to the

7 2^{\circ}

angle.

Solution

a Start by naming the islands so that it is easier to refer to them. Focus on

△ I U S

and ignore the map to have a clearer view.

Notice that $\overline{U B}$ is an extension of $\overline{I U}$ which means that $∠ B U S$ is an exterior angle to $△ I U S .$ Also, $∠ I$ and $∠ I S U$ are remote interior angles corresponding to $∠ B U S .$

The measure of

∠ B U S

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.

m ∠ B U S = m ∠ I + m ∠ I S U

Finally, substitute the corresponding measures to find the value of

x .

m ∠ B U S = m ∠ I + m ∠ I S U

SubstituteValues

Substitute values

x = 2 5^{\circ} + 1 3^{\circ}

AddTerms

Add terms

x = 3 8^{\circ}

b Focus on

△ S B U

this time.

In Part A, the measure of $∠ B U S$ is $3 8^{\circ} .$ The $7 2^{\circ}$ angle is an exterior angle to $△ S B U .$ Additionally, $∠ U$ and $∠ B$ are remote interior angles corresponding to the $7 2^{\circ}$ 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.

m ∠ B S T = m ∠ U + m ∠ B

Finally, substitute the measures of the angles and solve the equation for

y .

m ∠ B S T = m ∠ U + m ∠ B

SubstituteValues

Substitute values

7 2^{\circ} = 3 8^{\circ} + y

SubEqn

$LHS - 38 = RHS - 38$

3 4^{\circ} = y

RearrangeEqn

Rearrange equation

y = 3 4^{\circ}

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

5 5^{\circ} .

What is the value of

x ?

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

∠ A B D

is an exterior angle to

△ A B C .

Also,

∠ A

and

∠ C

are remote interior angles corresponding to

∠ A B D .

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.

m ∠ A B D = m ∠ A + m ∠ C

The measure of the exterior angle is

5 5^{\circ}

and the measures of the two remote interior angles were given in terms of

x .

Substitute the measure and expressions into the equation and solve it for

x .

m ∠ A B D = m ∠ A + m ∠ C

Substitute

$m ∠ A B D = 5 5^{\circ}$

5 5^{\circ} = m ∠ A + m ∠ C

SubstituteExpressions

Substitute expressions

5 5^{\circ} = 4 x + 3 (x + 2)

Distr

Distribute $3$

5 5^{\circ} = 4 x + 3 x + 6

AddTerms

Add terms

5 5^{\circ} = 7 x + 6

SubEqn

$LHS - 6 = RHS - 6$

4 9^{\circ} = 7 x

DivEqn

$LHS / 7 = RHS / 7$

7^{\circ} = x

RearrangeEqn

Rearrange equation

x = 7^{\circ}

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 $A B C .$ No matter the characteristics of this triangle, the sum of the measures of its interior angles is $18 0^{\circ} .$ 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.

m ∠ 4 m ∠ 5 m ∠ 6 = m ∠ 2 + m ∠ 3 = m ∠ 1 + m ∠ 3 = m ∠ 1 + m ∠ 2

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

18 0^{\circ} .

m ∠ 4 + m ∠ 5 + m ∠ 6 = (m ∠ 2 + m ∠ 3) + (m ∠ 1 + m ∠ 3) + (m ∠ 1 + m ∠ 2)

AddTerms

Add terms

m ∠ 4 + m ∠ 5 + m ∠ 6 = 2 m ∠ 1 + 2 m ∠ 2 + 2 m ∠ 3

FactorOut

Factor out $2$

m ∠ 4 + m ∠ 5 + m ∠ 6 = 2 (m ∠ 1 + m ∠ 2 + m ∠ 3)

Substitute

$m ∠ 1 + m ∠ 2 + m ∠ 3 = 18 0^{\circ}$

m ∠ 4 + m ∠ 5 + m ∠ 6 = 2 (18 0^{\circ})

Multiply

m ∠ 4 + m ∠ 5 + m ∠ 6 = 36 0^{\circ}

As shown, the sum of the measures of the exterior angles of a triangle, one at each vertex, is equal to

36 0^{\circ} .

Keep in mind that this result holds true for any triangle.

Level 1

Level 2

Level 3

Angles in Triangles

Exercises

Fill in the blank in the following sentences.

The sum of the measures of the interior angles of a triangle is $.$

Each exterior angle forms with its corresponding interior angle.

Given any triangle, the sum of the measures of the interior angles is equal to 180^(∘). This is what the Triangle Interior Angles Theorem says. This means that the missing part of the given statement is 180^(∘).

The sum of the measures of the interior angles of a triangle is 180^(∘).

We can justify this statement algebraically. Let's consider a triangle ABC.

Let's draw a line parallel to BC passing through A.

Notice that AC and AB are transversals that cut a pair of parallel lines. As a result, the alternate interior angles are congruent. This means that ∠ B ≅ ∠ DAB and ∠ C ≅ ∠ EAC.

The diagram shows that ∠ DAB, ∠ BAC, and ∠ EAC form a straight angle. This means that the sum of their measures is equal to 180^(∘). m∠ EAB + m∠ BAC + m∠ DAC = 180^(∘) We can substitute m∠ B for m∠ EAB and m∠ C for m∠ DAC. This is because congruent angles have the same measure. m∠ EAB+ m∠ BAC + m∠ DAC = 180^(∘) ⇓ m∠ B + m∠ BAC + m∠ C= 180^(∘) ✓ The last equation tells us that the sum of the measures of the interior angles of a triangle is equal to 180^(∘).

Let's draw a triangle and one of its exterior angles.

Next, let's consider the interior angle that is adjacent to ∠ SQR. In other words, let's consider ∠ PQR.

The diagram shows that the exterior angle and its corresponding interior angle form a linear pair, which means that their measures add up to 180^(∘). As a result, they form a straight angle. This is the phrase that fills in the blank!

Each exterior angle forms a straight angle with its corresponding interior angle.

Consider the following diagram.

Which of the labeled angles are exterior angles to the triangle? Select all that apply.

Consider the following diagram.

Which of the labeled angles are exterior angles to the triangle? Select all that apply.

Let's begin by recalling what an exterior angle of a triangle is.

An exterior angle is the angle formed between one side of a triangle and the extension of an adjacent side.

We can see that the sides of ∠ 2 are extensions of two sides of the triangle. In other words, none of the sides of ∠ 2 is a side of the triangle. The same situation happens with ∠ 3. This means that neither ∠ 2 nor ∠ 3 are exterior angles of the triangle.

On the other hand, we can see that ∠ 1, ∠ 4, and ∠ 5 all are formed by one side of the triangle and the extension of an adjacent side. As a result, we can conclude that ∠ 1, ∠ 4, and ∠ 5 are all exterior angles of the given triangle.

Notice that ∠ 3 and ∠ 5 are both formed by the extensions of the sides of the triangle. In other words, none of these angles is formed by one side of the triangle. This means that none of these angles is an exterior angle of the triangle.

We need to be careful about ∠ 1. As shown, it is formed by one side of the triangle and the extension of the same side. This is not what the definition says. It should be formed by one side of the triangle and the extension of an adjacent side to be an exterior angle. We then conclude that ∠ 1 is not an exterior angle.

On the contrary, ∠ 2, ∠ 4, and ∠ 6 are all formed by one side of the triangle and the extension of an adjacent side. This means that these three angles are exterior angles of the triangle.

For each of the following diagrams, identify the remote interior angles corresponding to the labeled exterior angle.

Select all that apply.

Let's begin by identifying the exterior angle of the triangle. An exterior angle is formed by one side of the triangle and an extension of an adjacent side. Also, this angle is outside the triangle. In the given diagram, ∠ 4 meets these conditions.

Now that we identified the exterior angle, let's recall the definition of a remote interior angle.

A remote interior angle is an interior angle that is not adjacent to a specific exterior angle.

Of the three interior angles of the triangle, ∠ 1 is adjacent to ∠ 4. This means that ∠ 1 is not a remote interior angle corresponding to ∠ 4. On the contrary, ∠ 2 and ∠ 3 meet the conditions in the definition and so, these two angles are remote interior angles corresponding to ∠ 4.

We can identify ∠ 1 as an exterior angle because it is outside the triangle and is formed by one side of the triangle and an extension of an adjacent side.

Recall, a remote interior angles is an interior angle that is not adjacent to a specific exterior angle. In our case, the remote interior angles corresponding to ∠ 1 are ∠ 2 and ∠ 4.

Find the missing angle measure in the following diagrams.

A triangle with a 37-degree and a 33-degree interior angles.

Find the measure of

∠ A .

A right triangle with a 40-degree angle.

Find the measure of

∠ L .

We are given a triangle and the measures of two interior angles. Our job is finding the third angle measure. Recall, the sum of the measures of the interior angles of a triangle is 180^(∘). Let's write an equation using this fact. m∠ A + m∠ B + m∠ C = 180^(∘) From the diagram, we have that m∠ B=33^(∘) and m∠ C = 37^(∘). Let's substitute these measures into the equation and solve it for m∠ A.

The missing angle measure is 110^(∘).

In △ JKL, we were given the measure of ∠ K and have to find the measure of ∠ L. This seems like insufficient information but, notice that ∠ J is a right angle which means that it measures 90^(∘).

Let's write an equation relating the three angle measures using the Interior Angles Theorem. From the obtained equation, we can find the missing angle measure.

Find the measure of the labeled exterior angle in each of the following diagrams.

A triangle with two interior angles with measures 35 and 25 degrees.

A right triangle with a 53-degree interior angle.

In △ MNO, the angles N and O are remote interior angles corresponding to the exterior angle PMN. Recall that there is a close relationship between these three angles.

Triangle Exterior Angle Theorem |- The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote angles.

Let's write an equation using this information. m∠ PMN = m∠ N + m∠ O Next, we substitute 25^(∘) for m∠ N and 35^(∘) for m∠ O. m∠ PMN &= 25^(∘) + 35^(∘) &⇓ m∠ PMN &= 60^(∘) Finally, let's update the given diagram and write the measure of the exterior angle.

Notice that ∠ EFG is an exterior angle, and ∠ D and ∠ E are its two corresponding remote interior angles.

We can write an equation relating the measures of these three angles using the Triangle Exterior Angle Theorem. m∠ EFG = m∠ D + m∠ E From the diagram, ∠ D is a right angle and so its measure is 90^(∘). Also, m∠ E = 53^(∘). Let's substitute these values into the equation to find the measure of the exterior angle. m∠ EFG &= 90^(∘) + 53^(∘) &⇓ m∠ EFG &= 143^(∘)

Angles in Triangles

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