3.1 Identify Pairs of Lines and 3.1 Identify Pairs of Lines and AnglesAngles

Objectives:

1.To differentiate between parallel, perpendicular, and skew lines

2.To compare Euclidean and Non-Euclidean geometries

VocabularyVocabulary

Parallel Lines Skew Lines

Perpendicular Lines

Euclidean Geometry

Transversal Alternate interior anglesAlternate

exterior angles

ConsecutiveInterior angles

ConsecutiveExterior angles

ConsecutiveExterior angles

Correspondingangles

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 1Example 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 LinesParallel Lines

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

The red arrows indicate that the lines are parallel.

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

Parallel LinesParallel Lines

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

Skew LinesSkew Lines

Example 2Example 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 2Example 2

3. Line(s) perpendicular to CD and containing point A.

4. Plane(s) parallel to plane EFG and containing point A.

AD

ABC

TransversalTransversal

A line is a transversaltransversal 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

TransversalTransversal

• <1 and <5 are corresponding anglescorresponding angles

• <3 and <6 are alternate alternate interior anglesinterior angles

• <1 and <8 are alternate alternate exterior anglesexterior angles

• <3 and <5 are consecutive interior consecutive interior anglesangles

Example 3Example 3

Classify the pair of numbered angles.

Corresponding

Alt. Ext

Alt. Int.

Example 4Example 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 5aExample 5a

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

Example 5bExample 5b

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

Perpendicular PostulatePerpendicular 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 PostulateParallel 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 PostulateEuclid’s Fifth Postulate

Euclid’s Fifth PostulateEuclid’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 6Example 6

If the Parallel Postulate is false, then what must be true?

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.

Example 6Example 6

If the Parallel Postulate is false, then what must be true?

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.

Example 6Example 6

If the Parallel Postulate is false, then what must be true?

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.

This is called a Poincare DiskPoincare Disk, and it is a 2D projection of a hyperboloidhyperboloid.

Example 6Example 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 6Example 6

If the Parallel Postulate is false, then what must be true?

2. Through a given point not on a given line, you can draw no line parallel to the given line. This makes Elliptic Geometry.

Example 6Example 6

This is a RiemannianRiemannian SphereSphere.

•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 GeometriesComparing Geometries

Parabolic Hyperbolic Elliptic

Also Known As

Euclidean GeometryLobachevskian

GeometryRiemannian Geometry

Geometric Model (Where Stuff Happens)

Flat Plane Poincare Disk* Riemannian Sphere

Comparing GeometriesComparing 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

Comparing GeometriesComparing Geometries

Parabolic Hyperbolic Elliptic

Curvature

NoneNegative

(curves inward, like a bowl)

Positive(curves outward, like a

ball)

Geometric Model (Where Stuff Happens)

Flat Plane Poincare Disk* Riemannian Sphere

Comparing GeometriesComparing Geometries

Parabolic Hyperbolic Elliptic

Applications

Architecture, building stuff (including pyramids,

great or otherwise)

Minkowski SpacetimeEinstein’s General

Relativity (Curved space)

Global navigation (pilots and such)

Geometric Model (Where Stuff Happens)

Flat Plane Poincare Disk* Riemannian Sphere

Great CirclesGreat 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 7Example 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 8Example 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 TrianglesCompare Triangles

Notice the difference in the sum in each picture