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: 1.To differentiate between parallel, perpendicular, and skew lines 2.To compare Euclidean and Non-Euclidean geometries

Vocabulary Parallel LinesSkew Lines Perpendicular Lines Euclidean Geometry TransversalAlternate interior angles Alternate exterior angles Consecutive Interior angles Consecutive Exterior angles Consecutive Exterior angles Corresponding angles In your notebook, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

Example 1 Use the diagram to answer the following. 1.Name a pair of lines that intersect. 2.Would JM and NR ever intersect? 3.Would JM and LQ ever intersect?

Parallel Lines 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. Parallel Lines

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

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? 1.Line(s) parallel to CD and containing point A. 2.Line(s) skew to CD and containing point A. AB AH

Example 2 3.Line(s) perpendicular to CD and containing point A. 4.Plane(s) parallel to plane EFG and containing point A. AD ABC

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

Example 3 Classify the pair of numbered angles. Corresponding Alt. Ext Alt. Int.

Example 4 List all possible answers. 1.<2 and ___ are corresponding <s 2.<4 and ___ are consecutive interior <s 3.<4 and ___ are alternate interior <s Answer in your notebook

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. Euclid’s Fifth Postulate 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? Hyperbolic Geometry Hyperbolic Geometry 1.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.Hyperbolic Geometry

Example 6 If the Parallel Postulate is false, then what must be true? Hyperbolic Geometry Hyperbolic Geometry 1.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.Hyperbolic Geometry

Example 6 If the Parallel Postulate is false, then what must be true? Hyperbolic Geometry Hyperbolic Geometry 1.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.Hyperbolic Geometry Poincare Disk hyperboloid This is called a Poincare Disk, and it is a 2D projection of a hyperboloid.

Example 6 DEFINITION: Parallel lines are infinite lines in the same plane that do not intersect. In the figure above, Hyperbolic Line BA and Hyperbolic Line BC are both infinite lines in the same plane. They intersect at point B and, therefore, they are NOT parallel Hyperbolic lines. Hyperbolic line DE and Hyperbolic Line BA are also both infinite lines in the same plane, and since they do not intersect, DE is parallel to BA. Likewise, Hyperbolic Line DE is also parallel to Hyperbolic Line BC. Now this is an odd thing since we know that in Euclidean geometry: If two lines are parallel to a third line, then the two lines are parallel to each other.

Example 6 If the Parallel Postulate is false, then what must be true? Elliptic Geometry Elliptic Geometry 2.Through a given point not on a given line, you can draw no line parallel to the given line. This makes Elliptic Geometry.Elliptic Geometry

Example 6 RiemannianSphere This is a Riemannian Sphere. Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. As all lines in elliptic geometry intersect

Comparing Geometries ParabolicHyperbolicElliptic Also Known As Euclidean Geometry Lobachevskian Geometry Riemannian Geometry Geometric Model (Where Stuff Happens) Flat PlanePoincare Disk*Riemannian Sphere

Comparing Geometries ParabolicHyperbolicElliptic 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 PlanePoincare Disk*Riemannian Sphere

Comparing Geometries ParabolicHyperbolicElliptic Curvature None Negative (curves inward, like a bowl) Positive (curves outward, like a ball) Geometric Model (Where Stuff Happens) Flat PlanePoincare Disk*Riemannian Sphere

Comparing Geometries ParabolicHyperbolicElliptic 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 PlanePoincare Disk*Riemannian Sphere

Great Circles Great Circle: 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. Great circles divide the sphere into two equal halves. l

Example 7 1.In Elliptic geometry, how many great circles can be drawn through any two points? 2.Suppose points A, B, and C are collinear in Elliptic geometry; that is, they lie on the same great circle. If the points appear in that order, which point is between the other two? Infinite Each is between the other 2

Example 8 For the property below from Euclidean geometry, write a corresponding statement for Elliptic geometry. For three collinear points, exactly one of them is between the other two. Each is between the other 2

Compare Triangles Notice the difference in the sum in each picture