1. More about Parallels

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

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

5. 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.

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

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

9. 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.

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

11. How big is the missing angle?

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

13. How big is the missing angle?

14. 180 – 36 – 57 = 87o

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

16. 1, 2, and 3 are all exterior angles.

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

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

20. = 100o

22. 40 + x = 100 … 100 – 40 = 60o

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

24. 5x + 13 = 4x x – 9 5x + 13 = 6x – 7

25. 5x + 13 = 6x – = x … 5x + 13 = 113o

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

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

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

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

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

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

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