Parallel Lines Cut by a Transversal

Now that people have GPS and navigation systems, it seems that fewer people really think through how to map out a route.  But maps can be a great way to show angle measures and patterns with parallel lines.  Reading a map is an important skill and following directions is a skill in itself, but I thought it would be fun to give directions using the geometry patterns and congruent angles made by parallel lines cut by a transversal.  Imagine if your navigation system used geometric prompts instead of just saying “In 200 feet, turn right”.  For example, the directions might be “Cross through the intersection to the vertical angle.”

So, I started by making a map of a town.  This town has to have parallel roads and then there’s a trolley line–which is the transversal–running through the center of town.  I have my students learn the terms while placing landmarks throughout the town.  Then students must become the navigation system and provide directions from various points.  The catch is that they cannot use directional terms like “turn right onto Apple Street,” instead, they must navigate through different streets and intersections by referring to the congruent angles and supplementary angles and by identifying the patterns found when parallel lines are cut by a transversal.

Set-up for Parallel Lines Cut by a Transversal

  1. Have an example of a map with parallel streets. Here is an NYC city map (note, this is not listed as public domain, but it links to a clear map online.  This can be used, or there is one below in Launch the activity section as well as on the worksheet)
  2. Provide each student with a blank map sheet found on the first page of the  Parallel Lines Cut by a Transversal Worksheet.
  3. Provide each student with a ruler.
  4. Provide each student with blue, red, and green colored pencils.
  5. Give each student one small piece of thin paper or sticky note.
  6. Students will work in pairs.
  7. Review parallel lines and demonstrate them using the map.
  8. Print out or have copies of the angle pairs and examples listed below (that are also attached to the worksheet).
  9. Have a large poster/paper or projection of the blank map to help students create their maps.
  10. The Parallel Lines Cut by a Transversal Worksheet provides various places the students must provide directions to travel between. For extra practice, students can write their own set of directions and see if their partner arrives at the desired location.

Launch the Parallel Lines Cut by a Transversal Activity

Students should have knowledge about angle sums and supplementary angles. However, by creating the map, students will review angle pairs and congruent angles. Show the students a map of New York City.  Ask them to look for parallel lines.

Here’s a map of Lower Manhattan showing parallel streets.

Maps are useful tools for demonstrating Parallel Lines Cut by a Transversal

Then tell them that you will be creating their own town map.  Introduce the blank map of parallel lines (found on page 2 of the worksheet). Ask the students how they know the lines are parallel.  They can use their rulers to help them out.

Then have the students line up the ruler along one of the streets and slowly slide the ruler up until it lines up with another street.  You can show them how each line is the same, they are just shifted up or down throughout the town.  Parallel lines will never cross each other.

Now tell the students to find the transversal.  For this map, we will make the transversal a trolley line running through town.  It is the double bold line that cuts through all of the parallel streets.  Ask them if this is parallel to the other streets.  Also, ask if it is perpendicular.  Note: The transversal is drawn at a different angle and is purposely not perpendicular for this activity.

Give the students time to name each street.  For consistency purposes, have them label the transversal as “Transverse Trolley Line”.

Then label the streets: North Apple Street, Center Street, and South Sunset Street.

One angle on the Parallel Lines Cut by a Transversal Worksheet map (page 2) is already labeled 46.  Have the students find the labeled angle. This is where they get to sketch an icon of their house.  On the worksheet, students will indicate their address.  The address must include the names of the 2 streets as well as the angle measure.

For example: On the corner of Transverse Trolley and North Apple Street, 46.

Have students take a sticky note (light yellow works well because they need to see the lines through the paper).  Place the edge of the sticky note along North Apple Street.  Trace over the line of the transversal and cut along that line.  This will create 2 supplementary angles.  Label the 46 angle and help the students figure out what the other angle would be:  180 – 46 = 134.  Tell the students to use the sticky note angles to help them prove congruency by moving them around and placing them at different angles.

Now tell the students that they will create their map, sketch each location, and label it on their worksheet.  Review the various examples of parallel lines cut by a transversal, and then have the students create their maps.

Here are the examples and terms.  Provide these examples for reference as the students try to follow directions in order to place locations on the map.  For example, students will be asked to “Place the Post Office at an alternate exterior angle from your house”.  They can use the templates below to figure out how to locate different types of angle pairs.  You can begin by placing some locations yourself, but try to provide examples and then let students label them on their own maps.  As they make their maps, walk around the room and check to make sure they are placing the examples correctly.

Here are the TERMS for angle pairs:

  • Vertical angles are congruent and across the intersection
  • Supplementary angles, add up to 180
  • Alternate interior angles are congruent – alternate sides of the transversal, and to the inside of the parallel lines
  • Alternate exterior angles are congruent – alternate side of the transversal and the exterior side of the parallel lines
  • Corresponding angles are congruent – on the same side of the transversal and on different parallel lines

Here is an example that shows each angle pair.  These are also located on the worksheet (p. 7,8)

  This public domain image has various angle pairs labeled

 examples of vertical angles

 Example of alternate exterior

 Example of alternate interior

Corresponding – same side of the transversal

 Supplementary Along a 180 line. (not congruent.  180-x and x)

 Consecutive angles – not congruent  (Same as supplementary)

Help students place the first location on the map to provide an example.  The instructions for placing the other locations are found on the Parallel Lines Cut by a Transversal worksheet.

The first instruction is “Place the Post Office at an alternate exterior angle from your house.” Show what “alternate exterior” means.  An alternate is indicating that the angles are on opposite or alternate sides of the transversal.  Exterior means that the angles lie to the outside of the parallel line pair.

Once students locate the Post Office, they can sketch it on the map.  Then ask the students to tell you what they think the angle measure is.  Alternate exterior angles are congruent, so if your house is at an angle of 46°, then the Post Office is at a congruent angle of 46° as well!

Now tell the students that they must work in pairs to complete the following three tasks:

  • First, they must work together to make sure they place the locations correctly by using the angle pairs and examples of parallel lines cut by a transversal.
  • Second, they must get their maps checked by the teacher or answer key.
  • Third, they will take turns giving directions from one point to another on the map. The points to travel to are found on the Parallel Lines Cut by a Transversal worksheet. The students must use the terms for angle pairs in order to navigate their partners to the right place.  The traveler will draw his or her path with a colored pencil.

Tell them that they must use the angle pair terms from the examples and avoid using navigation such as “turn left.”  They also cannot name the location until they have arrived.  Remind them that they can trace their path anywhere on a map, and they do not have to restrict the path along each road.

Here is an example of navigation steps from House to Supermarket :

  • First, walk from your house to the vertical angle. (park)
  • Then travel to the closest alternate interior angle, you will have to cross over the Transverse Trolley Line. (supermarket)

Map it Out and Navigate: Parallel Lines Cut by a Transversal Activity

Remind the students where to find the parallel lines cut by transversal angle pairs and terms.  These could be displayed at the front of the room or copied and provided at their seats.

Check that each group has the Post Office on the correct angle and periodically walk around to help them fill out their maps.  Locations and instructions are listed on the Parallel Lines Cut by a Transversal worksheet.

When a group thinks they are finished and ready to be checked, you can give them the answer sheet (worksheet p. 6) or check their map and give them the green light to move on to the navigation step.

Again, walk around and help the groups.  Have the groups write down the directions as well as say them out loud.  Some students may feel tempted to point to the places on the map, if this is the case or if you want to create a real life challenge, you could have the students provide step-by-step directions by ONLY writing, and no speaking!  Or the students can sit back to back, and look only at the map in front of them.  The student giving the directions must clearly say angle pairs and terms while the other student tries to trace the path on his/her map.

Make sure each student gets to provide the directions as well as navigate through the map.  There is one more set of points to travel between which can be done together or independently.  Once the path is traced, students must record the angle pair instructions.

Reflecting on Parallel Lines Cut by a Transversal Activity

This two-part activity helps students practice recognizing angle pairs and finding them on the map.  As you walk around, you can ask some of the following questions.  They can also be discussed at the end of the activity:

  1. Are vertical angles congruent?
  2. Can you find 2 sets of vertical angles where all 4 angles are congruent?
  3. Which types of angle pairs are not always congruent?
  4. Did you prefer following directions in order to navigate through the map of town or giving directions?  Why?
  5. Which angle pairs were the hardest for you to remember?
  6. What do the 4 angles add up to at any given intersection?
  7. Can you have alternate interior angles between South Sunset Street and North Apple Street?


  1. Create your own town with 6 parallel streets and 2 transversals.  Label locations and their angle pair associations.  Name your town and create a theme for your street names.  Place and draw at least 5 landmarks.  Be creative and colorful!
  2. Dance Dance Transversal – example video – This can be done by standing up and dancing.  First place tape of parallel lines and a transversal on the floor, (make sure it’s big enough for the students’ feet to fit) and move your feet to the correct positions according to the video.  Or students can tape or draw 2 parallel lines cut by a transversal and “dance” with their fingers. As they jump to the correct positions on the floor, have them call out the type of angles to help reinforce those terms.

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