What You Will Learn (Review) •

Points

•

Lines

•

Planes

•

Angles

9.1-1

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Basic Terms Three basic items in geometry: • Point • Line

9.1-2

•

plane are three basic terms in geometry

•

These three items are NOT given a formal definition

•

Yet we recognize them when we see them.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Lines, Rays, Line Segments

9.1-3

•

A line is a set of points.

•

Any two distinct points determine a unique line.

•

Any point on a line separates the line into three parts: the point and two half lines.

•

A ray is a half line including the endpoint.

•

A line segment is part of a line between two points, including the endpoints. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Basic Terms Description

Diagram

Line AB

A

Ray AB

B

B

A

Line segment AB

A

AB AB

B

A

Ray BA

9.1-4

Symbol

B

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

BA AB

Plane •

We can think of a plane as a two-dimensional surface that extends infinitely in both directions.

•

Any three points that are not on the same line (noncollinear points) determine a unique plane.

9.1-5

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane Two lines in the same plane that do not intersect are called parallel lines.

9.1-6

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane A line in a plane divides the plane into three parts, the line and two half planes.

9.1-7

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Plane •

Any line and a point not on the line determine a unique plane.

•

The intersection of two distinct, non-parallel planes is a line.

9.1-8

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles An angle is the union of two rays with a common endpoint; denoted by .

9.1-9

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Angles

9.1-10

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Types of Angles Adjacent Angles - angles that have a common vertex and a common side but no common interior points. Complementary Angles - two angles whose sum of their measures is 90 degrees. Supplementary Angles - two angles whose sum of their measures is 180 degrees. 9.1-11

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Problem solving

3x – 20 2

9.1-12

1

x

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Definitions When two straight lines intersect, the nonadjacent angles formed are called Vertical angles. Vertical angles have the same measure.

9.1-13

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Definitions A line that intersects two different lines, at two different points is called a transversal.

9.1-14

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Special Names (l1 and l2 are parallel)

9.1-15

Alternate interior angles 3 & 6; 4 & 5

Interior angles on the opposite side of the transversal–have the same measure

Alternate exterior angles 1 & 8; 2 & 7

Exterior angles on the opposite sides of the transversal–have the same measure

Corresponding angles 1 & 5, 2 & 6, 3 & 7, 4 & 8

One interior and one exterior angle on the same side of the transversal–have the same measure Copyright 2013, 2010, 2007, Pearson, Education, Inc.

1

2

3

4

5 6 7 8 1 3

2 4

5 6 7 8 1 3 5 6 7 8

2 4

Parallel Lines Cut by a Transversal When two parallel lines are cut by a transversal, 1. alternate interior angles have the same measure. 2. alternate exterior angles have the same measure. 3. corresponding angles have the same measure.

9.1-16

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 6: Determining Angle Measures The figure shows two parallel lines cut

by a transversal. Determine the

measure of through

9.1-17

.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

What You Will Learn •

Rigid Motion or Transformation

•

Reflections

•

Translations

•

Rotations

•

Glide Reflections

•

Tessellations

9.5-18

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Definitions The act of moving a geometric figure from some starting position to some ending position without altering its shape or size is called a rigid motion (or transformation).

9.5-19

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Reflection A reflection is a rigid motion that moves a geometric figure to a new position such that the figure in the new position is a mirror image of the figure in the starting position. In two dimensions, the figure and its mirror image are equidistant from a line called the reflection line or the axis of reflection. 9.5-20

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 1: Reflection of a Triangle Construct the reflection of triangle ABC about reflection line l.

B’

A’

C’

9.5-21

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Translation A translation (or glide) is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation.

9.5-22

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Translation A concise way to indicate the direction and the distance that a figure is moved during a translation is with a translation vector.

9.5-23

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 2: A Translated Square

Given square ABCD and translation vector v, construct the translated square A´B´C´D´.

9.5-24

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 2: A Translated Square

9.5-25

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Rotation A rotation is a rigid motion performed by rotating a geometric figure in the plane about a specific point, called the rotation point or the center of rotation. The angle through which the object is rotated is called the angle of rotation. 9.5-26

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: A Rotation Point Inside a Polygon

Given polygon ABCDEFGH and rotation point P, construct polygons that result from rotations through a) 90º

9.5-27

b) 180º.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: A Rotation Point Inside a Polygon (90: (x,y)  (-y,x)) a. 90 Solution: B

H’(-1,2)

C’(1, 4) D’(1, -2) E’(3, -2) F’(3, -4)

C A(2,3)

A’(-3,2) B’(-3,4)

4

A

H

G(-4,1) -4

2

H(2,1) -2

P

D(-2,-1) -2

F(-4,-3) E(-2,-3)

4

2

D

-4

G

G’(-1,-4) 9.5-28

B(4, 3)

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

E F

6

C(4, -1)

Example 5: A Rotation Point Inside a Polygon (180: (x,y)  (-x,-y)) a. 180 Solution: 4

H’(-2,-1)

A(2,3) E

A’(-2,-3) B’(-4,-3)

C’(-4,1) D’(2,1) E’(2,3)

2

G(-4,1) C -4

H(2,1) D -2

P

H D(-2,-1) -2

B A F(-4,-3) E(-2,-3)

2

4

6

G C(4, -1)

-4

F’(4,3) G’(4,-1) 9.5-29

B(4, 3) F

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Example 5: A Rotation Point Inside a Polygon

9.5-30

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Glide Reflection A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection.

9.5-31

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Tessellations A tessellation (or tiling) is a pattern consisting of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures used are called the tessellating shapes of the tessellation. 9.5-32

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

For Example The simplest tessellations use one single regular polygon.

9.5-33

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

For Example

9.5-34

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Creating a Tessellation with a Square Begin with a 2” square, draw a line. Cut and rotate.

9.5-35

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

What You Will Learn Non-Euclidean Geometry • Elliptical geometry • Hyperbolic geometry

9.7-36

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Euclid’s Fifth Postulate If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

9.7-37

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Euclid’s Fifth Postulate The sum of angles A and B is less than the sum of two right angles (180º). Therefore, the two lines will meet if extended.

9.7-38

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Playfair’s Postulate or Euclidean Parallel Postulate Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.

9.7-39

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Non-Euclidean Geometry •

Euclidean geometry is geometry in a plane.

•

Many attempts were made to prove the fifth postulate.

•

These attempts led to the study of geometry on the surface of a curved object:

- Hyperbolic geometry - Spherical, elliptical or Riemannian geometry

9.7-40

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Non-Euclidean Geometry A model may be considered a physical interpretation of the undefined terms that satisfies the axioms. A model may be a picture or an actual physical object.

9.7-41

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Fifth Axiom of Three Geometries Euclidean Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line. 9.7-42

Elliptical Given a line and a point not on the line, no line can be drawn through the given point parallel to the given line.

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Hyperbolic Given a line and a point not on the line, two or more lines can be drawn through the given point parallel to the given line.

A Model for the Three Geometries

The term line is undefined. It can be interpreted differently in different geometries. Euclidean

Elliptical

Plane

Sphere

Hyperbolic Pseudosphere 9.7-43

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Elliptical Geometry A circle on the surface of a sphere is called a great circle if it divides the sphere into two equal parts. We interpret a line to be a great circle. This shows the fifth axiom of elliptical geometry to be true. Two great circles on a sphere must intersect; hence, there can be no parallel lines. 9.7-44

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Elliptical Geometry If we were to construct a triangle on a sphere, the sum of its angles would be greater than 180º. The sum of the measures of the angles varies with the area of the triangle and gets closer to 180º as the area decreases. 9.7-45

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Hyperbolic Geometry Lines in hyperbolic geometry are represented by geodesics on the surface of a pseudosphere. A geodesic is the shortest and least-curved arc between two points on the surface. 9.7-46

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Hyperbolic Geometry This illustrates one example of the fifth axiom: through the given point, two lines are drawn parallel to the given line.

9.7-47

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Hyperbolic Geometry If we were to construct a triangle on a pseudosphere, the sum of the measures of the angles of the triangle would be less than 180º.

9.7-48

Copyright 2013, 2010, 2007, Pearson, Education, Inc.