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Consider when three lines are drawn. Eight angles are formed when two or fewer of those lines are parallel. The angles formed at one intersection point are related to the angles formed at the other. In the given diagram, there are no parallel lines.
Three lines intersecting at two different points.

This lesson will focus on finding relations between these eight angles under the condition that two lines are parallel.

Catch-Up and Review

Here is a recommended reading before getting started with this lesson.

Challenge

Fixing the Fence

It is a Friday afternoon. Dylan just arrive at home 🏠 from school 🏫. He begins the weekend by mending the fence in his backyard. It broke last night because of extremely strong wind 💨.

Fence

Things were going great. While nailing up some boards 🔨, he noticed something. All of the vertical boards are parallel. Also, the horizontal stringers make four angles with each vertical board. Which angles are congruent? Dylan wonders.

a Which angles are congruent to
b Which angles are congruent to
Discussion

Line Intersecting Two or More Lines

When a line intersects two lines at two different points, the line receives a special name.

Concept

Transversal

A transversal is a line that intersects two or more lines at different points.
Two lines and a Transversal
When a pair of lines are cut by a transversal, eight different angles are created. There are four angles around each point of intersection.
Eight angles formed by the transversal
Discussion

Angles Formed by a Transversal

The eight angles formed by a transversal are classified into different groups. These grouping are based on the angle positions relative to the lines involved. As an example, focus on the angles between the two lines and those that are not.

Concept

Interior and Exterior Angles

When two lines and are cut by a transversal eight different angles are created — four around each point of intersection. These angles are classified into two groups according to their positions with respect to lines and

  • Interior angles: the angles that are between the lines and
  • Exterior angles: the angles that are outside the lines and
Two lines and a transversal. Exterior and Interior regions highlighted.

In the diagram, all the even-numbered angles are interior angles, and all the odd-numbered angles are exterior angles.

Interior Angles Exterior Angles
Discussion

Relationships Between Interior Angles

Of the eight angles formed by a transversal that intersects two lines, the four interior angles are also classified according to their position with respect to the transversal.

Concept

Alternate Interior Angles

Consider a pair of lines cut by a transversal. The pairs of interior angles with different vertices that lie on opposite sides of the transversal are called alternate interior angles.
Alternate interior angles
In the diagram, two pairs of angles can be identified as alternate interior angles.
If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. The same logic in reverse can be applied. If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.
If Then
and
or
Example

The Louvre Museum

a Dylan really wants to makes art every minute of his free time. He finishes fixing the fence and runs to grab his canvas. It is just him and his canvas 🎨🖌️💗. He will continue painting a mosaic he began days ago. There are only eight regions left to be painted.
Mosaic
Of the eight regions, Dylan focuses on the two pairs of alternate interior angles. He wants to paint only the acute ones blue. What regions will Dylan paint blue? Select all that apply!
b Dylan's mosaic reminded his parents of some of the paintings they saw in the Louvre museum in Paris. They took out their photo album and Dylan excitedly flipped through it. Dylan saw the photo of the Louvre's entrance. Immediately, he could identify some parallel lines.
Louvre Entrance
External credits: Diego Lopez Sebastian
If lines and are parallel, what is the value of

Hint

a The transversal is the line that goes from the top edge to the bottom edge of the canvas. The alternate interior angles are the interior angles with different vertices that lie on opposite sides of the transversal. Remember that the angles must also be acute.
b When the lines are parallel, the alternate interior angles are congruent.

Solution

a Let's start by identifying which of the lines in the mosaic is the transversal. Due to the position of the regions, the transversal is the line that goes from the top edge to the bottom edge of the canvas. Let and be the lines that the transversal intersects.
Mosaic

The interior angles are those angles that lie between the two lines that are cut. In the mosaic, the regions and correspond to interior angles. Focus on these regions.

Mosaic
Alternate interior angles are interior angles with different vertices that lie on opposite sides of the transversal. Of the regions highlighted before, the following pairs correspond to alternate interior angles.
Dylan only wants the alternate interior angles that are acute. The regions and correspond to acute angles. The regions and are obtuse angles. These can be verified with the help of a protractor. Dylan is now ready to paint the regions and blue.
Mosaic

Nicely done! Now Dylan has six regions remaining to paint.

b The line intersects the lines and Therefore, it is a transversal. Now, ignore the photo and focus on the three lines to have a clear picture of the lines. Also, label some points on the lines.
Louvre Entrance
The angles and are interior angles with different vertices that lie on opposite sides of the transversal. Therefore, these two angles are alternate interior angles.
Since the lines and are parallel, the mentioned angles are congruent. As a result, they have the same measure. That is enough information for the value of to be identified.

Extra

Interesting Facts About the Louvre
  • It is the most visited art museum in the world.
  • It is the largest museum in the world. It covers over acres.
  • Originally, it was constructed as a fortress by King Phillip of France in In the century, the Louvre was turned into a royal residence. Since it was opened to the public as Muséum central des arts de la République.
  • From to it was named Napoleon Museum.
  • The Mona Lisa was stolen from the Louvre in This incident rose the popularity of the painting across the globe. It was recovered two years later.
  • During World War II, the Louvre was emptied to protect all of the art pieces.
  • There are five pyramids in the Louvre.
  • In the second Louvre of the world was opened. It is located in Abu Dhabi.
Louvre Birds-eye
Discussion

Relationships Between Exterior Angles

It was just taught that the four interior angles can be divided into two pairs of alternate interior angles. The same can be said for exterior angles. They can also be grouped into two pairs.

Concept

Alternate Exterior Angles

Consider a pair of lines cut by a transversal. The pairs of exterior angles with different vertices that lie on opposite sides of the transversal are called alternate exterior angles.
Alternate exterior angles
In the diagram, two pairs of angles can be identified as alternate exterior angles.
If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. The same logic in reverse can be applied. If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.
If Then
and
or

So far, the eight angles formed by a transversal and two parallel lines have been divided into two groups. In addition, each of these two groups has been divided into a subgroup.

Mind Map
Example

La Plata City - Argentina

Dylan continued to flip through the photo book. A picture his parents took in La Plata, Argentina fascinated him. He did an internet search to learn more. He discovered that it is one of the best-planned cities of the world. It is shaped like a square grid with some diagonal avenues, squares, and traffic circles.

<credit>Gindelis</credit>

Astonished by the structure of the city, he zoomed in on the map and saw that many interior streets are parallel to each other. For example, he noticed that the streets named Calle and Calle are parallel.

Map of La Plata, Argentina
a Assume has a measure of and has a measure of What is the value of
b Assume and What is the value of

Hint

a Use the fact that the given angles are alternate exterior angles. Also, the streets are parallel to each other. Note that the names of the streets are shown in its original form of Spanish. The diagrams of the Solution also show translations to English. The word calle in Spanish means street in English.
b Notice that streets named Calle and Calle are parallel to each other. The involved angles are alternate exterior angles. Note that the names of the streets are shown in its original form of Spanish. The diagrams of the Solution also show translations to English. The word calle in Spanish means street in English.

Solution

a It is important to have a clear picture of the mentioned angles. Draw the angles on the map to get started.
Map of La Plata, Argentina

Notice that and lie over the street named Diagonal This implies that these two segments lie over the same line. Also, lies over the street named Calle and lies over Calle That means the lines containing the segments are parallel since the two streets are parallel.

Map of La Plata, Argentina

The diagram shows two parallel lines cut by a transversal. This means that and are alternate exterior angles.

Two parallel lines cut by a transversal
The alternate exterior angles are congruent because the lines and are parallel. That means they have the same measure. An equation in terms of will be set and solved using this information.
b Again, it is important to have a clear vision of the angles drawn on the map.
Map of La Plata, Argentina

Note that and lie over the street named Diagonal That means these two segments lie over the same line. Additionally, lies over Calle and lies over Calle These two streets are parallel. This implies that the lines containing the segments are also parallel.

Map of La Plata, Argentina

The diagram shows two parallel lines cut by a transversal. That characteristic means that and are alternate exterior angles.

Two parallel lines cut by a transversal
The alternate exterior angles are congruent because the lines and are parallel. This means that Additionally, because they have the same measure. Set and solve an equation in terms of using what information is now known.
Discussion

Interior Angles on the Same Side of the Transversal

The alternate interior angles lie on opposite sides of the transversal. Now it is time to get to know the interior angles with different vertices that lie on the same side of the transversal.

Concept

Same-Side Interior Angles

Consider a pair of lines cut by a transversal. The pairs of interior angles with different vertices that lie on the same side of the transversal are called same-side interior angles or consecutive interior angles.
Same-side interior angles
Alternatively, same-side interior angles are called co-interior angles. In the diagram, two pairs of same-side interior angles can be identified.
If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary. The same logic in reverse can be applied. If two lines and a transversal form consecutive interior angles that are supplementary, then the lines are parallel.
If Then
and
or
Example

Pyramids of Giza

Dylan is still mesmerized by the travel album. He is staring at a photo of the Giza Pyramids in Egypt. His parents teach him about them. They were built as tombs of Egyptian pharaohs. The tallest pyramid is called the Great Pyramid. There are seven wonders of the ancient world — only the pyramids are still standing.

Pyramids of Giza
Dylan's father says that the angle at the top of the Great Pyramid is about The non-common sides of the marked angles are parallel to each other. What is the value of

Hint

Draw the sides of the angles marked in the photo. Notice that these angles are consecutive interior angles.

Solution

Start by drawing the sides of the angles marked in the photo to relate these two angles.

Pyramids of Giza

Notice that the non-common sides of the angles are parallel to each other. Additionally, the common side can be seen as a transversal. The marked angles lie between the parallel lines and they are on the same side of the transversal.

Pyramids of Giza; Two parallel lines, a transversal, and a pair of consecutive interior angles
The angle's characteristics suggest that they are consecutive interior angles. Additionally, the two angles are supplementary because the two lines are parallel. This means that the sum of their measures is equal to The following equation can be set using the given information.
Finally, solve the obtained equation to find the value of

Extra

Seven Wonders of the Ancient World
  • The Pyramids of Giza
  • Hanging Gardens of Babylon
  • Temple of Artemis
  • Statue of Zeus
  • Mausoleum at Halicarnassus
  • Colossus of Rhodes
  • Lighthouse of Alexandria
Discussion

Exterior Angles on the Same Side of the Transversal

It is now time to learn about the grouping of exterior angles that have different vertices but are on the same side of the transversal.

Concept

Same-Side Exterior Angles

Consider a pair of lines cut by a transversal. The pairs of exterior angles with different vertices that lie on the same side of the transversal are called same-side exterior angles or co-exterior angles.
Same-side exterior angles
In the diagram, two pairs of same-side exterior angles can be identified.
If two parallel lines are cut by a transversal, then the same-side exterior angles are supplementary. The same logic in reverse can be applied. If two lines and a transversal form same-side exterior angles that are supplementary, then the lines are parallel.
If Then
and
or
Example

Number of Countries in Africa

Dylan was loving the stories his parents were telling. He asked for them to tell another. One of his parents looked through the photos and saw an elephant 🐘. What a beautiful safari they had while in Zimbabwe.

Landscape of Zimbabwe

Dylan's curiosity peaked. He asked for more details about Zimbabwe. His parents told him that it is a country in South East Africa whose flag is made up of many parallel lines. The flag reminds them of a riddle they were challenged with during the trip. Dylan's parent drew some angles on the flag to prepare the riddle.

Flag of Zimbabwe

His parent begins the riddle.

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

Join Dylan and solve the riddle.

Hint

Draw the sides of the angles marked on the flag. The horizontal sides are parallel to each other. The other sides can be connected to form a transversal. Notice that the marked angles are same-side exterior angles.

Solution

Try to relate the two marked angles in a way that the value of can be found. It seems like a good idea to start by drawing the sides of these angles.

Flag of Zimbabwe

Notice that the horizontal sides are parallel to each other. Additionally, the sides lying on the border of the triangular region lie on the same line. For this reason, those two sides can be connected to form a transversal.

Flag of Zimbabwe

Dylan's parent drew the angles on top of the flag. Draw the angles now without the flag for a clearer view.

Two parallel lines and a transversal
Notice that the angles lie on the exterior of two parallel lines and on the same side of the transversal. This means the angles are same-side exterior angles. These two angles are supplementary and their measures add up to because of that.
Solve the previous equation to determine the value of
The value of is Finally, review the main part of the riddle once more.
Since the number of countries in Africa is times Performing this multiplication gives There are countries in Africa.
Discussion

Relating Interior and Exterior Angles

When a transversal cuts two lines, the interior angles can be paired in two different ways: as alternate angles or same-side angles. Something similar happens with exterior angles. Notice that each of these pairs is made up of either two interior angles or two exterior angles.

Two parallel lines cut by a transversal

It is worth asking if an interior angle can be related to an exterior angle. The following concept establishes that type of relationship.

Concept

Corresponding Angles

Consider a pair of lines cut by a transversal. The pairs of angles with different vertices that lie on the same position relative to the vertex are called corresponding angles. In the diagram, and are corresponding angles because they are both northeast of their vertices.
Corresponding angles
Alternatively, two angles are corresponding angles if they meet the following conditions.
  • One angle is exterior and the other is interior.
  • The angles have different vertices.
  • The angles lie on the same side of the transversal.

In the diagram, four pairs of angles can be identified as corresponding angles.

Pair Position Relative to the Vertex Position Relative to the Transversal
and Northeast Right
and Northwest Left
and Southwest Left
and Southeast Right

If two parallel lines are cut by a transversal, then the corresponding angles are congruent. The same logic in reverse can be applied. If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

If Then
and
or
Example

Bridges and Boats

Dylan is having an incredible weekend full of stories and learning. He is impressed with all the countries his parents have visited. So far, they have talked about France, Argentina, Egypt, and Zimbabwe. With a huge smile, Dylan's parents tell him that they have been to Japan too.

World-Map-with-some-Countries.png

In Japan they sailed ⛵ along the longest suspension bridge in the world at the time. It is called Akashi Kaikyo Bridge. In disbelief, Dylan looked for those photos in the album.

a Dylan began to recognize things about the bridge. He notices how some of the diagonal beams of the bridge footings are parallel.
Akashi Kaikyo Bridge
What is the measure of the angle labeled below the bridge?
b Next, Dylan checks out a photo of their sailboat. He notices something in the middle of the sails. There are two bars parallel to each other.
Akashi Kaikyo Bridge
What is the measure of the angle labeled at the top of the sail?

Hint

a Draw the sides of the angles labeled. The vertical sides of each angle lie over the vertical beam. As a result of that, they lie on the same line. The vertical beam acts like a transversal. Notice that the angles are corresponding angles because they lie on the same position relative to their vertices.
b Both angles are northwest to their vertices. Note these angles are corresponding angles.

Solution

a Try to find a relationship about the given angles to find the measure of the lower angle. It is a good idea to begin by drawing the sides of each angle.
Akashi Kaikyo Bridge

Notice that the vertical sides lie on the left vertical beam. That implies they lie on the same line. This line acts like a transversal cutting the two parallel diagonal beams.

Akashi Kaikyo Bridge

Draw the given information without the photo in the background for a clearer visual of the angles, lines, and transversal.

Two parallel lines cut by a transversal
The diagram shows that the angles lie on the same side of the transversal. One of them is an exterior angle while the other is an interior angle. This means that these angles are corresponding angles. The angles have the same measure because the lines are parallel.
Next, solve the equation for
Now that the value of is known, substitute it into the measure of the lower angle.
The measure of the angle labeled below the bridge is Notice that the angle labeled above the bridge also measures
b Start by drawing the lines containing the sides of the angles labeled.
Akashi Kaikyo Bridge

The two diagonal bars are parallel to each other according to what Dylan sees. The upper angle is exterior while the lower angle is an interior angle. Additionally, both angles lie on the same side of the transversal.

Two parallel lines cut by a transversal
Those characteristics suggest that these two angles are corresponding angles. As a result, they have the same measure.