1. Warm-Up
The graph shown represents a hyperbola. Draw the solid of revolution formed by rotating the hyperbola around the y-axis.

3. Warmer-Upper
The image shown is a print by M.C. Escher called Circle Limit III. Pretend you are one of the golden fish toward the center of the image. What you think it means about the surface you are on if the other golden fish on the same white curve are the same size as you?

5. 3.1 Identify Pairs of Lines and Angles
Objectives:
To differentiate between parallel, perpendicular, and skew lines
To compare Euclidean and Non-Euclidean geometries

6. Vocabulary
As a group, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.
Parallel Lines
Skew Lines
Perpendicular Lines
Euclidean Geometry
Transversal

7. Example 1 Use the diagram to answer the following.
Name a pair of lines that intersect.
Would JM and NR ever intersect?
Would JM and LQ ever intersect?

8. Parallel Lines
Two lines are parallel lines if and only if they are coplanar and never intersect.
The red arrows indicate that the lines are parallel.

9. Parallel Lines
Two lines are parallel lines if and only if they are coplanar and never intersect.

10. Skew Lines
Two lines are skew lines if and only if they are not coplanar and never intersect.

11. Example 2
Think of each segment in the figure as part of a line. Which line or plane in the figure appear to fit the description?
Line(s) parallel to CD and containing point A.
Line(s) skew to CD and containing point A.

12. Example 2 Line(s) perpendicular to CD and containing point A.
Plane(s) parallel plane EFG and containing point A.

13. Transversal
A line is a transversal if and only if it intersects two or more coplanar lines.
When a transversal cuts two coplanar lines, it creates 8 angles, pairs of which have special names

14. Transversal <1 and <5 are corresponding angles
<3 and <6 are alternate interior angles
<1 and <8 are alternate exterior angles
<3 and <5 are consecutive interior angles

15. Example 3
Classify the pair of numbered angles.

16. Example 4 List all possible answers.
<2 and ___ are corresponding <s
<4 and ___ are consecutive interior <s
<4 and ___ are alternate interior <s

17. Example 5a
Draw line l and point P. How many lines can you draw through point P that are perpendicular to line l?

18. Example 5b
Draw line l and point P. How many lines can you draw through point P that are parallel to line l?

19. Perpendicular Postulate
If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

20. Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Also referred to as Euclid’s Fifth Postulate

21. Euclid’s Fifth Postulate
Some mathematicians believed that the fifth postulate was not a postulate at all, that it was provable. So they assumed it was false and tried to find something that contradicted a basic geometric truth.

22. Example 6 If the Parallel Postulate is false, then what must be true?
3.1 and 3.2
Example 6
If the Parallel Postulate is false, then what must be true?
Through a given point not on a given line, you can draw more than one line parallel to the given line. This makes Hyperbolic Geometry.
Add links to GSP files that demonstrate each kind of geometry.

23. Example 6 If the Parallel Postulate is false, then what must be true?
3.1 and 3.2
Example 6
If the Parallel Postulate is false, then what must be true?
Through a given point not on a given line, you can draw more than one line parallel to the given line. This makes Hyperbolic Geometry.
Add links to GSP files that demonstrate each kind of geometry.

24. Example 6 If the Parallel Postulate is false, then what must be true?
3.1 and 3.2
Example 6
If the Parallel Postulate is false, then what must be true?
Through a given point not on a given line, you can draw more than one line parallel to the given line. This makes Hyperbolic Geometry.
Add links to GSP files that demonstrate each kind of geometry.
This is called a Poincare Disk, and it is a 2D projection of a hyperboloid.

25. Example 6 If the Parallel Postulate is false, then what must be true?
3.1 and 3.2
Example 6
If the Parallel Postulate is false, then what must be true?
Through a given point not on a given line, you can draw more than one line parallel to the given line. This makes Hyperbolic Geometry.
Add links to GSP files that demonstrate each kind of geometry.
Click the hyperboloid.

26. Example 6 If the Parallel Postulate is false, then what must be true?
3.1 and 3.2
Example 6
If the Parallel Postulate is false, then what must be true?
Through a given point not on a given line, you can draw no line parallel to the given line. This makes Elliptic Geometry.
Add links to GSP files that demonstrate each kind of geometry.

27. Example 6 If the Parallel Postulate is false, then what must be true?
3.1 and 3.2
Example 6
If the Parallel Postulate is false, then what must be true?
Through a given point not on a given line, you can draw no line parallel to the given line. This makes Elliptic Geometry.
Add links to GSP files that demonstrate each kind of geometry.
This is a Riemannian Sphere.

28. Comparing Geometries Parabolic Hyperbolic Elliptic Also Known As
Euclidean Geometry
Lobachevskian Geometry
Riemannian Geometry
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere

29. Comparing Geometries Parabolic Hyperbolic Elliptic
Parallel Postulate: Point P is not on line l
There is one line through P that is parallel to line l.
There are many lines through P that are parallel to line l.
There are no lines through P that are parallel to line l.
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere

30. Geometric Model (Where Stuff Happens)
Comparing Geometries
Parabolic
Hyperbolic
Elliptic
Curvature
None
Negative
Positive
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere

31. Comparing Geometries Parabolic Hyperbolic Elliptic Applications
Architecture, building stuff (including pyramids, great or otherwise)
Minkowski Spacetime
Einstein’s General Relativity (Curved space)
Global navigation (pilots and such)
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere

32. Great Circles
Great Circle: The intersection of the sphere and a plane that cuts through its center.
Think of the equator or the Prime Meridian
The lines in Euclidean geometry are considered great circles in elliptic geometry. l
Great circles divide the sphere into two equal halves.