# parallel lines

## PARALLEL LINES CAN MEET?

Parallel lines run side-by-side, in the same direction.

How could they ever meet?

Think of train tracks.

Do they ever meet?

### A riddle

Share this riddle with family and friends.

• Zork is camping.
• She steps out of her tent.
• She walks south 1 km.
• She walks west 1 km.
• She sees a bear and gets scared.
• She runs north 1 km.
• … and arrives back at her tent!
• How is that possible?
• And, what colour was the bear?

Here’s a hint: look at this globe of the Earth.

If parallel lines never meet, then the Earth is flat, like a wall map.

### Euclid

Parallel lines have a really interesting history!

The ancient Greek mathematician Euclid stated a parallel lines theorem that went something like this:

• Draw a straight line, and a point near it.
• There exists exactly 1 line that crosses the point and does not cross the first line.
• The two lines are parallel.

Click on the image below to see a 3-D animation of our interview with Euclid (who claimed that parallel lines never meet)

### What geometry do you live in?

What do you think of Euclid’s theorem?

How many “straight” lines can you draw through the point that don’t cross the line:

1. no lines?
2. one line?
3. an infinite number of lines?

Surprise! The answer depends on the geometry you have in mind.

Below are images of 3 different geometries: Euclidean, Spherical and Hyperbolic. Can you match each of these with the three answers above?

### The shortest path

One way to define a “straight” line is to say that it is the “shortest” path between two points.

Which lines on the globe create the shortest paths?

• the lines of latitude?
• the lines of longitude?
• some other lines?

Imagine a bug walking “straight ahead” on the globe.

• What path will it travel?
• Could it be a line of latitude?
• Could it be a line of longitude?
• Could it be a different line?

Let’s consider an airplane flying from Tokyo to New York City. Would it fly along a line of latitude? Or would it take a different route?

### Triangles on a flat surface

What is the sum of the angles in a triangle?

If you draw a triangle, cut off its 3 angles, and then put them together to form a combined angle, that combined angle would be 180 degrees.

### Triangles on a sphere

We can draw a triangle on a sphere using 3 great circles (the equivalent of straight lines), such as a two lines of longitude and the equator, as shown below.

What is the sum of the angles in the triangle shown above?

### What did you learn?

What did you learn about parallel lines?

• What surprised you?
• What math insights did you experience?

Share your learning with others, so they can experience math surprise and insight as well?