Chapter 1 Tools of Geometry

1-1 Patterns and Inductive Reasoning

1-2 Points, Lines, and Planes

1-3 Segments , Rays, Parallel Lines, and Planes

1-4 Measuring Segments and Angles

1-5 Basic Constructions

1-6 The Coordinate Plane

1-7 Perimeter, Circumference, and Area

Definitions• Acute angle• Angle bisector• Collinear points• Congruent angles• Congruent segments• Conjecture• Coordinate• Coplanar• Counterexample• Inductive reasoning• Line• Obtuse angle• Parallel lines• Parallel planes

• Perpendicular bisector• Perpendicular lines• Plane• Point• Postulate• Ray• Right angle• Segment• Skew lines• Space• Straight angle

1-1 Patterns and Inductive Reasoning

• Goal : To use inductive reasoning to make conjectures

Example 1Use inductive reasoning to find a pattern for each sequence. Use the pattern to show the next two terms in the sequence.

a. 3,6,12,24,…

3 6 12 24

2 x 24 = 48 and 2 x 48 = 96

b.

Each circle has one more segment through the center of the circle.

Inductive Reasoning

• Reasoning based on patterns you observe.

A, E, I, ?, ?

Make a conjecture about the sum of the first 50 odd numbers.

1

1 + 3

1 + 3 + 5

1 + 3 + 5 + 7

= 1

= 4

= 9

= 16

What are these numbers?Perfect squares!!!

=

=

=

=

What can we conclude if we were to keep going?

The sum of the first 30 odd numbers is 30 squared or 900

24

21

22

23

Conjecture

• A conclusion reached by inductive reasoning

Example: Can you make a conjecture about the next Batman movie based on clues from The Dark Knight?

What characters may be in it? Why?

You are using both inductive reasoning and making a conjecture!

The speed with which a cricket chirps is affected by the temperature. If you hear 20 cricket chirps in 14 seconds, what is the temperature?

5 chirps 45°F

10 chirps 55°F

15 chirps 65°F

Chirps per 14 Seconds

75°FJeff works out regularly. When he first start exercising, he could do 10 push-ups. After the first month he could do 14. After the second month he could do 19, and after the third month he could do 25. Predict the number of push-ups Jeff will be able to do after the fifth month of working out. How confident are you of your prediction? Explain.

40Time and maximum number are two factors in making the prediction only an estimate.

Find one counter example to show that the conjecture is false.

•The difference of two integers is less than either integer.

-6 – (-4) < -6 False-6 – (-4) < -4 False

Since neither is true this is a counterexample.

Homework• p. 6 – 7: 1, 12,17,21,22,25,28,57,58

tB

A

1-2 Points, Lines, and Planes

Point

Space

Line

• Goal- To understand basic terms and postulates of geometry.

Point

• A location• Has no size• Represented by a small dot and is named by a

capital letter.

A

Space

• The set of all points

Line

• A series of points that extends in two opposite directions with no end.

• A line is named by any two points on the line, such as (read “line AB”) or by a single lower case letter such as t ( read “line t”).

AB

n

Identifying Collinear Pointsa. Are points E,F, and C collinear? If so, name the line they lie on.

b. Are points E,F, and D collinear? If so, name the line they lie on.

m

l

C

D

EF

P

a. Points E,F, and D are collinear. They lie on line m.

b. Points E,F, and D are not collinear.

Collinear Points

• Points that lie on the same line.• Collinear rays would be rays that lie on the

same line.

X

In the figure below, name three points that are collinear and three points that are not collinear.

Y

W

Z

V

P

Plane P

Plane ABC

C

BA

coplanar

Plane

• A flat surface that has no thickness.• A plane contains many lines and extends

without end in the directions of all its lines.• You name a plane by either a single capital

letter (that is not a point on the plane) or by AT LEAST three of its noncollinear points.

Noncollinear

• Points that DO NOT lie on the same plane.

Coplanar

• Points and lines that are in the same plane.

Naming a Plane

a. What do the dotted lines in a picture like this one mean?

b. Each surface of the ice cube represents part of a plane. Name the plane represented by the front of the ice cube.

c. List three different names for the plane represented by the top of the ice cube.

D

HG

F

E

A

B

C

a. The object is 3D, and the dotted lines represent the sides you cannot see.

b. Plane AEF, Plane AEB, and Plane ABFE (these are only a few of the names).

c. Plane HEF, Plane HEFG, and Plane FGH (these are only a few of the names).

Homework

• P 13-14

1-2 Points, Lines, and Planes Part 2

A postulate or axiom is an accepted statement or fact.

Postulate 1-1

Through any two points there is exactly one line.

A

B

t

• Line t is the only line that passes through points A and B.

• What does that mean for any two points?

A

H

F

E

DC

B

Any two points make exactly one line!

Postulate 1-2

If two lines intersect, then they intersect in exactly one point.

A

B

C

D E

AE and

intersect at C.

BD

Postulate 1-3

If two planes intersect, then they intersect in exactly one line.

RW

S

T

Plane RST and plane STW intersect at

ST

a. What is the intersection of plane HGFE and plane BCGF?

b. Name two planes that intersect at BF

H

E

G

F

A B

CD

F

a. FG

b. Plane ABF and Plane CBF (can be named in other ways)

Postulate 1-4

Through any three noncollinear points there is exactly one plane.

Using Postulate 1-4Shade the plane that contains A,B, and C.

E

H G

F

A B

CD

Shade the plane that contains E,H, and C.

E

H G

F

A B

CD

Homework

• P 13 – 15: 2, 11, 31, 44, 45

1-3 Segments, Rays, Parallel Lines, and Planes

• Goal- To identify segments and rays.• Goal- To recognize parallel lines.

A B

Endpoints

X

Y

RQ S

Segment ABRay XY

Opposite Rays

Segment

• The part of a line consisting of two endpoints and all the points between them.

• You name a segment by it’s endpoints with a line over it. Ex. AB

Ray

• The part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.

• The endpoint must be the first letter!!!

A

B

This is ray AB

Opposite Rays

• Two collinear rays with the same endpoint.• Opposite rays ALWAYS form a line.

Naming Segments and Rays

Name the segments and rays in the figure.

L

P

Q

Parallel Lines

• Coplanar lines that do not intersect.

Skew Lines

• Noncoplanar, therefore, they ARE NOT parallel and DO NOT intersect.

E

H G

F

A B

CD

AB EFand are Parallel or Skew Lines?

AB CGand are Parallel or Skew Lines?

Parallel Planes

• Planes that do not intersect.

In the classroom what planes are parallel?

Can you think of other parallel planes?

Homework

• P 19 – 22 # 5, 16,17,40,41,50,54,55-60,75,81,83

1-4 Measuring Segments and Angles

Goal:• To find the lengths of segments• To find the measures of angles

Ruler Postulate

• The distance between any two points is the absolute value of the difference of the two numbers.

0 2 4 6 8-8 -6 -4 -2

Length of AB

42 AB = 6

= 6

Congruent Segments

• We show two things are congruent with the following symbol:

A B

C D

2cm

2cm

A B

C D

AB = CD CDAB

0 1 2 3 4 5 6 7 8-8 -7 -6 -5 -4 -3 -2 -1

A B C D

AC =

DB =

Segment Addition Postulate

A

B

C

AB + BC = AC

A

B

C

a. If AB = 5 and BC = 9, then AC = ???

b. If BC = 7 and AC = 20, then AB = ???

A

B

C

4x-20

2x+3

AC = 100

C is the midpoint of . Find AC, CB, and AB.AB

A C B2x + 1 3x - 4

Midpoint

• A point that divides a segment into two congruent segments.

Angle

• Formed by 2 rays with the same endpoint.• Rays are the sides of the angle. The endpoint

is the vertex of the angle.• Expressed by:

Naming an Angle X

A

B

1

Vertex

Possible names for the angle: X, 1, or (with the vertex in the middle)

AXB, and BXA

One way to measure an angle is in degrees. To indicate the size or degree of an angle write a lowercase m in front of the angle symbol.

78°P

Q

X

m X = 78°Name angle X in two other ways:

Classifying Angles

Acute Angle0 < x < 90

x°x°

Right Angle x = 90

x°

Obtuse Angle90 < x < 180

x°

Straight AngleX = 180

Angle Addition Postulate

O

AB

C

m AOC + m BOC = m AOC O

A

B

C

m AOC + m BOC = 180

Using the Angle Addition Postulate

What is m TSW if m RST = 50 and m RSW = 125

SR

T W

Marking Congruent Angles

A

X

A B

Homework

• P 25-28 # 5,10,14,31-33,34,37,48,49

1-5 Basic Constructions

A few things we need to know:• Perpendicular Lines• Perpendicular Bisector•Angle Bisector

Perpendicular Lines

• Two lines that intersect to form right angles.• The symbol means “ is perpendicular to”

Perpendicular Bisector

• A line, segment, or ray that is perpendicular to a segment at its midpoint.

Angle Bisector

• A ray that divides an angle into two angles.

1-6 The Coordinate Plane

Goal: To find the distance between two points in the coordinate plane.Goal: To find the coordinate of the midpoint of a segment in the coordinate plane.

(x,y) describes the location of a point.

Quadrant I(+, +)

Quadrant II(-, +)

Quadrant III(-, -)

Quadrant IV(+, -)

J

K

L

M

The Distance Formula

Given points and then:

),( 11 yxA ),( 22 yxB

212

212 )()( yyxxd

Memorize THIS!!!!!!!!

Finding DistanceFind the distance between T(5,2) and R(-4,-1) to the nearest tenth.

Every morning Jordan takes the “Blue Line” subway from Sycamore Station to Byron Station. Sycamore Station is 1 mile West and 2 miles South of Emerald City. Byron Station is 2 miles East and 4 miles North of Emerald City. Find the distance Jordan travels between Sycamore Station and Byron Station?

Byron Station

Sycamore StationEmerald City

Midpoint FormulaThe coordinates of the MIDPOINT M of a segment with endpoints

and are:),( 11 yxA ),( 22 yxB

)2

,2

( 2121 yyxxM

MEMORIZE THIS!!!

has endpoints Q(3,5) and S(7,-9). Find the coordinates of its midpoint M.QS

The midpoint of is M(3,4). One endpoint is A(-3,-2). Find the coordinates of the other endpoint B.

AB

Homework

• 1-5 P 33-36 # 8-10, 14, 15, 17, 35, 39, 40

• 1-6 P 40-42 # 2, 5, 9, 12, 16, 19, 31, 32,46

1-7 Perimeter, Circumference, and Area

Goal: • To find the perimeters of rectangles and squares,

and circumferences of circles.• To find areas of rectangles, squares, and circles.

PerimeterThe sum of the lengths of all the sides of a polygon.

++

++

+

+

+

+

+

+

++

+

++

+

+

+

PerimeterSQUARE

RECTANGLE

b

h

b

h

P = 2b + 2h

dr

O

C = Circumferenced = diameterr = radius

C = d

C = 2r

CIRCLE

Your pool is 15 ft wide and 20 ft long with a 3 ft wide deck surrounding it. You want to build a fence around the deck. How much fencing will you need?

3 ft

20 ft

15 ft

Finding CircumferenceFind the circumference of circle A in terms of . Then find the circumference to the nearest tenth.

A

12 in

Y

X

A

BC

Find the Perimeter of ABC.

You are designing a banner for homecoming. The size of the banner will be 6 ft wide and 8 ft high. How much material do you need?

The diameter of a circle is 10 in. Find the area in terms of .

Homework

• P 43 Project ( steps 1-5), P47 # 3,14 • P 56- 59 #7, 12, 26, 42, 49, 52, 64, 66

Postulate 1-9

• If two figures are , then their areas are equal.

Postulate 1-10

• The area of a region is the sum of the areas of its non-overlapping parts.

3 m

10 m

12 m

4 m

2 m

Find the Perimeter and Area of the figure.

Homework

• P 55 16-19, 41(a &b), 42

Homework

• P 65 -68 # 7, 17, 29, 51, 58, 66, 71, 72

END OF CHAPTER!!!!!• Make sure to Review P 71-73 mult of 3 • Complete the given study guide as well