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

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?

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

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

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?

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.

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

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

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.

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

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

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

Example 3 Classify the pair of numbered angles.

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

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

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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.