Chapter One

Getting Started…

Inductive Reasoning

Making conclusions/predictions based on patterns and examples. Find the next two terms:

3, 9, 27, 81, . . .Draw the next picture:

Find the next two terms:384, 192, 96, 48, . . .

243, 729

24, 12

Making a Conjecture

Make a conclusion based on inductive reasoning.

Use the table to make a conjecture about the sum of the first six positive even numbers.

2 = 2 = 1·22 + 4 = 6 = 2·32 + 4 + 6 = 12 = 3·42 + 4 + 6 + 8 = 20 = 4·52 + 4 + 6 + 8 + 10 = 30 = 5·6

= 6·7 = 42

Counterexample

(like a contradiction) An example for which the conjecture is incorrect.

Conjecture: the product of two positive numbers is greater than either number.

6

1

3

1

2

1

111 counterexample

Fun Patterns

Find the next character in the sequenceJ, F, M, A, . . .January, February, March, April, MayFind the next character in the sequenceS, M, T, W, . . .Sunday, Monday, Tuesday, Wednesday,

ThursdayFind the next character in the sequenceZ, O, T, T, F, F, S, S, . . .Zero, One, Two, Three, Four, Five, Six, Seven, EightFind the next character in the sequence3, 3, 5, 4, 4, . . .One has 3 letters, Two has 3, Three has 5, Four has 4, Five has 4, Six has 3

Lesson 1-1

Points, Lines, and PlanesEssential Understandings:

The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.

Essential Questions:How do algebraic concepts relate to geometric concepts?How do patterns and functions help us represent data and solve real-word problems?

Points A point names a

location and has no size. It is represented by a dot.

A

B AC

Always use a CAPITAL letter to name a point.

Never name two points with the same letter (in the same sketch).

Lines A straight path that

has no thickness and extends forever.

m

AB

C

ABCEFFFFFFFFFFFFF F

Use a lowercase italicized letter or two points on the line.

Never name a line using three points.

Collinear PointsCollinear points are points that lie on the same line. (The line does not have to be visible.)

A B C DF

E

Collinear Non collinear

Planes

A plane is a flat surface that has no thickness and extends forever.

KB

E

Plane R, or IKE, KEB, BIE, BKE, IKE, KIE, etc.

Usually represented by a rectangle or parallelogram.

Use an italicized CAPITAL letter or any three non-collinear points. (Sometimes four are used.)

R

I

CANNOT name BIK as these points are collinear.

Different planes in a figure:A B

CD

EF

GH

Plane ABCD

Plane EFGH

Plane BCGF

Plane ADHE

Plane ABFE

Plane CDHG

Etc.

Coplanar Objects

Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

H

E

G

DC

BA

F

Are the following points coplanar?

A, B, C ?A, B, C, F ?H, G, F, E ?E, H, C, B ?A, G, F ?C, B, F, H ?

YesNoYesYesYesNo

Postulate

An accepted statement or fact.

Postulate 1-1

Through any two points there is exactly one line.(Say what? Two points make a line.)

l

B

A

Through any three noncollinear points there is exactly one plane. (Say what? Three non-collinear points make a plane.)

H

E

G

DC

BA

F

Plane AFGD

Plane ACGE

Plane ACH

Plane AGF

Plane BDG

Etc.

Postulate 1-2

Postulate 1-3

If two points lie on a plane, then the line containing those points lies on the plane.

BA

P

Postulate 1-4

If two lines intersect, then they intersect in exactly one point. (Say what? Two lines intersect at a point.)

B

A

C

D

P

P

R

A

B

Plane P and Plane R intersect at the line AB

EFFFFFFFFFFFFF F

Postulate 1-5If two planes intersect, then they intersect in exactly one line. (Say what? Two planes intersect in a line.)

3 Possibilities of Intersection of a Line and a Plane

(1) Line passes through plane — intersection is a point.(2) Line lies on the plane — intersection is a line.(3) Line is parallel to the plane — no common points.

Segments (line segments)

Part of a line consisting of two points (endpoints) and all the points inbetween.

Q

R

Use the two endpoints to name a segment.

Do not show the endpoints in the name.

Rays

Part of a line that starts at an endpoint and extends forever in one direction.

Use the endpoint as the first letter and any other point on the ray.

AY

The arrow in the name always goes left to right regardless of the physical ray.

Opposite rays form a line and share an endpoint.

X

G FE

Parallel/Skew and Coplanar/Non-Coplanar

H

E

G

DC

BA

F

Parallel planes are planes that do not intersect.

Parallel lines are coplanar lines that do not intersect.

Skew lines are non-coplanar lines which do not intersect.

FE

GH

A

C

B

D

Congruence and Tick Marks

Congruent segments are segments that have the same length.

Tick marks are used in diagrams to show congruence.

Q

P

SR

PQ and RS represent numbers. Use equality for numbers.

vs

and represent geometric figures. Use congruence for figures.

Segment Bisector

The midpoint of a segment bisects the segment into two congruent segments.

MAC

A segment bisector is a ray, segment, or line that intersects a segment at a midpoint.

MAC

Segment Addition Postulate

If B is between A and C, then

A B C

Ruler Postulate

If the coordinates of points A and B are the numbers a and b, then the distance AB is written as:

A B