More about Parallels

First, a side note …. We’ll start this section by First, a side note … We’ll start this section by talking about triangles.

You probably already know that there are 180o in a triangle.

In order to explain this, we need to start with an important rule called the parallel postulate. This is the most controversial of Euclid’s five original postulates.

So there always is a parallel line through a point, and there’s only one parallel line.

The idea is that any other line you draw through the point will eventually intersect the original line.

This is a postulate. It can’t be proved This is a postulate. It can’t be proved. However on flat surfaces it does appear to be obvious.

Once you know this, it’s easy to show a triangle has 180o.

How big is the missing angle?

How big is the missing angle? 180 – 40 – 70 = 70o

How big is the missing angle?

180 – 36 – 57 = 87o

Exterior Angle An angle formed by extending one of the sides of a triangle

1, 2, and 3 are all exterior angles.

Remote Interior Angles. . Inside the triangle. . Not adjacent to the Remote Interior Angles  Inside the triangle  Not adjacent to the exterior angle

Exterior Angle Theorem The measure of an exterior angle is equal to the sum of the remote interior angles.

45 + 55 = 100o

40 + x = 100 … 100 – 40 = 60o

Solve for “x”, and find the exterior angle.

5x + 13 = 4x + 2 + 2x – 9 5x + 13 = 6x – 7

5x + 13 = 6x – 7 20 = x … 5x + 13 = 113o

Important The properties we know about triangles rely on the parallel postulate. They work fine on flat surfaces.

If a surface is curved, though, strange things happen.

For instance, on the surface of the earth every triangle has more than 180o.

A triangle drawn on the bell of a brass instrument will have less than 180o.

There are called non-Euclidean geometries There are called non-Euclidean geometries. They vary, depending on what we call a plane.

Non-Euclidean geometries are used in situations like navigating over long distances.

 Parallel postulate  180o in a triangle  Exterior angle theorem  Non-Euclidean geometries