Angles and Transversals: Alternate, Corresponding, and Interior Angles

Interior Angles	Exterior Angles
$∠ 2,$ $∠ 4,$ $∠ 6,$ $∠ 8$	$∠ 1,$ $∠ 3,$ $∠ 5,$ $∠ 7$

Interior Angles

Exterior Angles

∠ 2,

∠ 4,

∠ 6,

∠ 8

∠ 1,

∠ 3,

∠ 5,

∠ 7

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 3 ≅ ∠ 5$ and $∠ 4 ≅ ∠ 6$
$∠ 3 ≅ ∠ 5$ or $∠ 4 ≅ ∠ 6$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 3 ≅ ∠ 5

and

∠ 4 ≅ ∠ 6

∠ 3 ≅ ∠ 5

∠ 4 ≅ ∠ 6

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 1 ≅ ∠ 7$ and $∠ 2 ≅ ∠ 8$
$∠ 1 ≅ ∠ 7$ or $∠ 2 ≅ ∠ 8$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 1 ≅ ∠ 7

and

∠ 2 ≅ ∠ 8

∠ 1 ≅ ∠ 7

∠ 2 ≅ ∠ 8

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$m ∠ 3 + m ∠ 6 = 18 0^{\circ}$ and $m ∠ 4 + m ∠ 5 = 18 0^{\circ}$
$m ∠ 3 + m ∠ 6 = 18 0^{\circ}$ or $m ∠ 4 + m ∠ 5 = 18 0^{\circ}$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

m ∠ 3 + m ∠ 6 = 18 0^{\circ}

and

m ∠ 4 + m ∠ 5 = 18 0^{\circ}

m ∠ 3 + m ∠ 6 = 18 0^{\circ}

m ∠ 4 + m ∠ 5 = 18 0^{\circ}

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$m ∠ 1 + m ∠ 8 = 18 0^{\circ}$ and $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$
$m ∠ 1 + m ∠ 8 = 18 0^{\circ}$ or $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

m ∠ 1 + m ∠ 8 = 18 0^{\circ}

and

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

m ∠ 1 + m ∠ 8 = 18 0^{\circ}

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

ℓ_{1} ∥ ℓ_{2}

Consider what I drew on Zimbabwe's flag. The number of countries in all of Africa equals three times $x .$ How many countries are there in Africa?

Pair	Position Relative to the Vertex	Position Relative to the Transversal
$∠ 1$ and $∠ 5$	Northeast	Right
$∠ 2$ and $∠ 6$	Northwest	Left
$∠ 3$ and $∠ 7$	Southwest	Left
$∠ 4$ and $∠ 8$	Southeast	Right

Pair

Position Relative to the Vertex

Position Relative to the Transversal

∠ 1

and

∠ 5

Northeast

Right

∠ 2

and

∠ 6

Northwest

Left

∠ 3

and

∠ 7

Southwest

Left

∠ 4

and

∠ 8

Southeast

Right

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 1 ≅ ∠ 5,$ $∠ 2 ≅ ∠ 6,$ $∠ 3 ≅ ∠ 7,$ and $∠ 4 ≅ ∠ 8$
$∠ 1 ≅ ∠ 5,$ $∠ 2 ≅ ∠ 6,$ $∠ 3 ≅ ∠ 7,$ or $∠ 4 ≅ ∠ 8$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 1 ≅ ∠ 5,

∠ 2 ≅ ∠ 6,

∠ 3 ≅ ∠ 7,

and

∠ 4 ≅ ∠ 8

∠ 1 ≅ ∠ 5,

∠ 2 ≅ ∠ 6,

∠ 3 ≅ ∠ 7,

∠ 4 ≅ ∠ 8

ℓ_{1} ∥ ℓ_{2}

Angles	Relation
$∠ 7$ and $∠ 4$	Alternate Interior Angles
$∠ 7$ and $∠ 3$	Consecutive Interior Angles
$∠ 7$ and $∠ 2$	Corresponding Angles
$∠ 7$ and $∠ 5$	Vertical Angles

Angles

Relation

∠ 7

and

∠ 4

Alternate Interior Angles

∠ 7

and

∠ 3

Consecutive Interior Angles

∠ 7

and

∠ 2

Corresponding Angles

∠ 7

and

∠ 5

Vertical Angles

Angles	Relation	Relationship Between Measures
$∠ 7$ and $∠ 4$	Alternate Interior Angles	$Congruent$
$∠ 7$ and $∠ 3$	Consecutive Interior Angles	$Supplementary$
$∠ 7$ and $∠ 2$	Corresponding Angles	$Congruent$
$∠ 7$ and $∠ 5$	Vertical Angles	$Congruent$

Angles

Relation

Relationship Between Measures

∠ 7

and

∠ 4

Alternate Interior Angles

Congruent

∠ 7

and

∠ 3

Consecutive Interior Angles

Supplementary

∠ 7

and

∠ 2

Corresponding Angles

Congruent

∠ 7

and

∠ 5

Vertical Angles

Congruent

Angles	Relation	Relationship Between Measures
$∠ 1$ and $∠ 8$	Alternate Exterior Angles	$Congruent$
$∠ 1$ and $∠ 5$	Same-Side Exterior Angles	$Supplementary$
$∠ 1$ and $∠ 6$	Corresponding Angles	$Congruent$
$∠ 1$ and $∠ 3$	Vertical Angles	$Congruent$

Angles

Relation

Relationship Between Measures

∠ 1

and

∠ 8

Alternate Exterior Angles

Congruent

∠ 1

and

∠ 5

Same-Side Exterior Angles

Supplementary

∠ 1

and

∠ 6

Corresponding Angles

Congruent

∠ 1

and

∠ 3

Vertical Angles

Congruent

Classify the marked angles according to their positions for each of the following diagrams.

Two lines cut by a transversal and two angles labeled.

The angles the lines form with the transversal can be classified although the lines are not parallel. Here, the two marked angles are between l_1 and l_2. This means that the angles are interior angles.

We can also see that they are on opposite sides of the transversal.

Let's sum up what we know. The two angles are interior angles with different vertices that lie on opposite sides of the transversal. These characteristics allow for us to say that the marked angles are alternate interior angles.

Notice that both angles are outside the lines l_1 and l_2. This means that the marked angles are exterior angles.

These two angles are on the same side of the transversal. That is different than what we found in Part A.

The two marked angles are exterior angles with different vertices that are on the same side of the transversal. We conclude that the angles are same-side exterior angles.

One of the first things we can see is that the lines are parallel. Be aware that this does not change the way of classifying the angles. This time, we can see that one angle is between the lines and the other is outside them.

We can also see that the angles are in the same position relative to their vertices — both are southeast of their vertices. This means that the marked angles are corresponding angles.

Commercial airplanes usually take off at an angle between $5^{\circ}$ to $1 5^{\circ} .$ The exact angle depends on the airplane and the sky conditions. This morning, Cameron's plane took off at an angle of $1 5^{\circ} .$ A few seconds passed. It then crossed a trail made by a plane that flew by earlier.

The trail left by the other plane is parallel to the airport runway. What is the value of

x ?

Let's begin by making a diagram that illustrates the described situation. Remember, the trail left by the other plane and the airport runway are parallel.

The plane's path is a transversal that cuts the old trail and the airport runway. Also, the two labeled angles are between the horizontal lines and on the same side of the transversal. That means the two angles are same-side interior angles. The two angles are supplementary since the old trail and the runway are parallel. x + 15^(∘) &= 180^(∘) &⇓ x &= 165^(∘) We conclude that the angle formed by the plane's path and the trail left by the other plane has a measure of 165^(∘).

Consider the following diagram. Lines $ℓ_{1}$ and $ℓ_{2}$ are parallel.

Two parallel lines, a transversal, and a ray

What is the measure of

∠ H D G ?

In the given diagram, notice that the required angle along with ∠ FDG and ∠ FDE form a straight angle.

The sum of the measures of these three angles is equal to 180^(∘). m∠ HDG + m∠ FDG + m∠ FDE = 180^(∘) We might wonder why we set the equation above if we do not know the measure of ∠ FDE. Well, notice that ∠ FDE is northwest of D, and ∠ ABD is also northwest of B. This implies that these two angles are corresponding angles. Corresponding Angles ∠ FDE & ∠ ABD The corresponding angles are congruent because lines l_1 and l_2 are parallel and t is a transversal. For this reason, we conclude that ∠ FDE and ∠ ABD have the same measure. m∠ FDE &= m∠ ABD &⇓ m∠ FDE &= 40^(∘) Next, let's substitute the measure of ∠ FDE into the equation written at the beginning. Then, we will solve it for m∠ HDG. Recall that the diagrams shows m∠ FDG=60^(∘).

We calculated the measure of the required angle to be 80^(∘). There is one last thing to do before finishing. Let's update the diagram by adding the measures we found.

Consider the following diagram where $A B$ is parallel to $D E .$

Two parallel lines and one angle between them.

What is the value of

x ?

Notice that the labeled angles are not formed by the same transversal. Then, we cannot relate them directly. However, we can try to find a connection between them by drawing an auxiliary line. Draw a line that passes through C and is parallel to AB and DE. Also, consider a point F on this line.

The diagram shows that CF divides the angle ACD into two angles, ∠ ACF and ∠ FCD. Therefore, the sum of the measures of these two new angles is equal to x. x = m∠ ACF + m∠ FCD That method caused us to have two more unknown angles! Why did we do this? Well, we can relate these two new angles to the two given angle measures. Begin by focusing on DE, CF, and the angles formed by the transversal CD.

The angles FCD and CDE are between the lines DE and CF. They are also on opposite sides of the transversal. These are the characteristics of alternate interior angles. Even more, they are congruent since the lines are parallel. As a result, they have the same measure. ∠ FCD &≅ ∠ CDE &⇓ m∠ FCD &= m∠ CDE In the given diagram, we can see that m∠ CDE=30^(∘). Then, m∠ FCD=30^(∘). Let's repeat the same reasoning but focus on the two lines below and the transversal AC.

The angles BAC and ACF are between the lines and on the same side of the transversal. For this reason, they are consecutive interior angles. The consecutive interior angles are supplementary because AB and CF are parallel, . Supplementary Angles ∠ ACF & ∠ BAC The sum of their measures is equal to 180^(∘) according to the definition of supplementary angles. m∠ ACF + m∠ BAC = 180^(∘) We see that m∠ BAC = 135^(∘). Let's substitute this into the equation to find the measure of ∠ ACF.

We now know the measures of ∠ ACF and ∠ FCD. They can be substituted into the first equation we wrote. This will determine the value of x.

The measure of ∠ ACD is 75^(∘).

Determine whether the lines $ℓ_{1}$ and $ℓ_{2}$ are parallel to each other.

Notice that the two labeled angles are between the lines l_1 and l_2. They are interior angles.

Additionally, the angles are on the same side of the transversal. This means that the angles are consecutive interior angles. Let's recall a fact about a situation just like this which will help us determine if the lines are parallel.

If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.

Now, we will verify whether the labeled angles are supplementary. Let's add their measures. 100^(∘) + 81^(∘) = 181^(∘) The sum is not equal to 180^(∘). This means that the angles are not supplementary. As a result of that, the lines are not parallel.

Notice that the two labeled angles are outside the lines l_1 and l_2. They are exterior angles.

In addition, the angles are on opposite sides of the transversal. This means that the angles are alternate exterior angles. Next, let's recall a fact about this type of situation which will help us determine whether the lines are parallel.

If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.

The two labeled angles have the same measure according to the diagram. That means the angles are congruent. Then, we can conclude that the lines are parallel.

Consider the following diagram.

Assume the measure of

∠ 3

5 0^{\circ} .

What is the measure of

∠ 7 ?

Assume the measure of

∠ 4

12 0^{\circ} .

What the measure of

∠ 8 ?

Assume the measure of

∠ 6

12 5^{\circ} .

What is the measure of

∠ 5 ?

Notice that ∠ 3 and ∠ 7 have different vertices and lie in the same position relative to their vertices. Both are west of their vertices. This means that these two angles are corresponding angles.

Additionally, the lines r and s have triangular hatch marks. This means that they are parallel. Now, remember that when two parallel lines are cut by a transversal, the corresponding angles are congruent. In other words, corresponding angles have the same measure. ∠ 3 ≅ ∠ 7 ⇓ m∠ 3 = m∠ 7 Since we were told that the measure of ∠ 3 is 50^(∘), we can say that the measure of ∠ 7 is also 50^(∘).

This time, let's relate ∠ 4 and ∠ 8. Take a look at the diagram. They have different vertices and lie on the exterior of lines r and s. This means that both are exterior angles. Also, they are on the same side of the transversal. As a result, they are same-side exterior angles.

As in Part A, we note that the lines r and s are parallel. Remember that when two parallel lines are cut by a transversal, the same-side exterior angles are supplementary. This means that the measures of the angles add up to 180^(∘). m∠ 4 + m∠ 8 = 180^(∘) We are told that the measure of ∠ 4 is 120^(∘). Let's substitute it into the equation and solve it for m∠ 8.

Finally, let's relate ∠ 6 and ∠ 5. Notice that both have different vertices and lie between the lines. They are interior angles. Additionally, both are on opposite sides of the transversal. For this reason, we conclude that the two angles are alternate interior angles.

As before, remember that the lines r and s are parallel. Even more, when two parallel lines are cut by a transversal, the alternate interior angles are congruent. In other words, alternate interior angles have the same measure. ∠ 6 ≅ ∠ 5 ⇓ m∠ 6 = m∠ 5 Since the measure of ∠ 6 is 125^(∘), we conclude that the measure of ∠ 5 is also 125^(∘).

Angles and Transversals

Catch-Up and Review

Fixing the Fence

Line Intersecting Two or More Lines

Transversal

Angles Formed by a Transversal

Interior and Exterior Angles

Relationships Between Interior Angles

Alternate Interior Angles

The Louvre Museum

Hint

Solution

Extra

Relationships Between Exterior Angles

Alternate Exterior Angles

La Plata City - Argentina

Hint

Solution

Interior Angles on the Same Side of the Transversal

Same-Side Interior Angles

Pyramids of Giza

Hint

Solution

Extra

Exterior Angles on the Same Side of the Transversal

Same-Side Exterior Angles

Number of Countries in Africa

Hint

Solution

Relating Interior and Exterior Angles

Corresponding Angles

Bridges and Boats

Hint

Solution

Extra

Identifying Pairs of Angles

Parallel Lines and Transversals

Angles and Transversals

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