Relationships Between Angles: Interior and Exterior Angles of a Triangle

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

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}

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 $∠ 6$	$m ∠ 5 + m ∠ 9 = 18 0^{\circ}$ $m ∠ 6 + m ∠ 9 = 18 0^{\circ}$

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}

Consider the following diagram.

Triangle ABC. Angle A measures 126 degrees and the exterior angle at C measures 150 degrees.

Find the measure of

∠ B .

Consider the following diagram.

Triangle FGH. Angle H measures 34 degrees and the exterior angle at G measures 90 degrees.

What is the measure of

∠ F ?

We start by noticing that ∠ A and ∠ B are remote interior angles corresponding to the exterior angle BCD. Recall the following fact that relates their measures.

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.

We can then write an equation using this information. m∠ BCD = m∠ A + m∠ B From the diagram, m∠ BCD = 150^(∘) and m∠ A = 126^(∘). Let's substitute these values into the equation and solve it for m∠ B.

As shown, the measure of ∠ B is 24^(∘).

We begin by noticing two main things about the given triangle.

The exterior angle FGI is a right angle and so, its measure is 90^(∘).
The angles H and F are remote interior angles corresponding to ∠ FGI.

The second statement along with the Triangle Exterior Angle Theorem allows us to write the following equation. m∠ FGI = m∠ H + m∠ F As said before, the measure of the exterior angle is 90^(∘). Also, the measure of ∠ H is 34^(∘). Let's substitute these values and solve the equation for m∠ F.

Our calculations show that ∠ F has a measure of 56^(∘).

Consider the following diagram.

What is the value of

x ?

We can see that x is the measure of ∠ DBC which is an interior angle of △ DBC. For that triangle, it is a given that the measure of ∠ C is 15^(∘).

Remember, the measures of the interior angles of a triangle add up to 180^(∘). Let's apply this fact to △ DBC.

It became our mission to find the measure of ∠ BDC. According to the diagram, this angle and ∠ EDF are vertical angles. That means they have the same measure — vertical angles are congruent. Then, it is enough to find the measure of ∠ EDF. With this in mind, let's focus on △ DEF.

We have that ∠ EFA is an exterior angle to this triangle. This means that m∠ EFA is equal to the sum of the measures of its two remote interior angles. m∠ EFA = m∠ E + m∠ EDF Notice that the exterior angle is a right angle which means that it has a measure of 90^(∘). Also, we can see that m∠ E = 18^(∘). Let's substitute these values into the previous equation.

Now that we know the measure of ∠ EDF, we can say that the measure of ∠ BDC is 72^(∘). Remember, ∠ EDF and ∠ BDC have the same measure. Finally, let's substitute 72^(∘) into the first equation we wrote to find the value of x. x &= 165^(∘) - 72^(∘) &⇓ x &= 93^(∘)

Consider the following diagram.

Find the value of

m ∠ B + m ∠ D .

Notice that ∠ DAE is a right angle and so, it has a measure of 90^(∘). Also, this angle is inside △ ADE.

We can use the fact that the measures of the interior angles of a triangle add up to 180^(∘) to write an equation and find the measure of ∠ D.

We also have that ∠ DAE is an exterior angle corresponding to △ ABC.

Remember, the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. We can find the measure of ∠ B using this information.

We already found the needed angle measures. Finally, we can calculate their sum. m∠ B + m∠ D &= 43^(∘) + 35^(∘) &⇓ m∠ B + m∠ D &= 78^(∘)

Angles in Triangles

Catch-Up and Review

Investigating the Sum of the Angle Measures

Interior Angles of a Triangle

Relationship Between the Interior Angles of a Triangle

Interior Angles Theorem

Bermuda Triangle

Hint

Solution

A Starry Night

Hint

Solution

Angles Outside a Triangle

Exterior Angles of a Triangle

Remote Interior Angles

Relationship Between an Exterior Angle and Its Remote Interior Angles

Triangle Exterior Angle Theorem

Beach's Day

Hint

Solution

Angles in the Ship

Hint

Solution

Finding the Missing Angle Measure

Relationship Between the Exterior Angles of a Triangle

Angles in Triangles

Recommended exercises

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