**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:

- no lines?
- one line?
- 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?