§ 1.1 Patterns and Inductive Reasoning Patterns and Inductive ReasoningPatterns and Inductive Reasoning § 1.4 Conditional Statements and Their Converses

Reasoning in GeometryReasoning in GeometryReasoning in GeometryReasoning in Geometry

§§ 1.1 1.1 Patterns and Inductive Reasoning

§§ 1.4 1.4 Conditional Statements and Their Converses

§§ 1.3 1.3 Postulates

§§ 1.2 1.2 Points, Lines, and Planes

§§ 1.6 1.6 A Plan for Problem Solving

§§ 1.5 1.5 Tools of the Trade

2) Solve the equation. Check your answer.

5 Minute-Check5 Minute-Check

1) Both answers can be calculated. Which one is right? What makes it right? What makes the other one incorrect?

30632 X 20632 X

243314 xx

3) If a dart is thrown at the circle to the right, what is the probability that it will land in a yellow sector? The odds?

total

favorableP

eunfavorabl

favorableO

Find the value or values of the variable that makes each equation true.

1.

2.

3.

4.

5.

6. Find the next three terms of the sequence. 6, 12, 24, . . .

3 63g

12 7 67x 22 32y

2 4 3 6 0z z

If 4 and 3, what is the value of the expression

2 5 3 ?

c d

d c

g = 21

x = 5

y = 4 or y = - 4

z = - 2

6

48, 96, 192


Patterns and Inductive ReasoningPatterns and Inductive Reasoning

If you were to see dark, towering cloudsapproaching, you might want to take cover.

Your past experience tells you that athunderstorm is likely to happen.

When you make a conclusion based on a pattern of examples or past events, you are using inductive reasoning.

You will learn to identify patterns and use inductive reasoning.


You can use inductive reasoning to find the next terms in a sequence.

Find the next three terms of the sequence:

3, 6, 12,

X 2X 2

24, 48, 96,

X 2 X 2 X 2

Find the next three terms of the sequence:

7, 8, 11,

+ 3+ 1

16,

+ 5 + 7 + 9

3223,


Draw the next figure in the pattern.


A _________ is a conclusion that you reach based on inductive reasoning.

In the following activity, you will make a conjecture about rectangles.

conjecture

1) Draw several rectangles on your grid paper.

2) Draw the diagonals by connecting each corner with its opposite corner. Then measure the diagonals of each rectangle.

3) Record your data in a table

d1 = 7.5 in.

d2 = 7.5 in.

Diagonal 1 Diagonal 2

Rectangle 1 7.5 inches 7.5 inches

Make a conjecture about the diagonals of a rectangle


A conjecture is an educated guess.

Sometimes it may be true, and other times it may be false.

How do you know whether a conjecture is true or false?

Try different examples to test the conjecture.

If you find one example that does not follow the conjecture, then the conjecture is false.

Such a false example is called a _____________.counterexample

Conjecture: The sum of two numbers is always greater than either number.

Is the conjecture TRUE or FALSE ?

Counterexample: -5 + 3 = - 2 - 2 is not greater than 3.



Find the next three terms of each sequence.

1.

2.

3.

4.

5.5, 6.5, 8.5, 11.5, ...

59, 63, 67

15.5, 20.5, 26.5

-2 + 4 = 2 and 2 < 4

Draw the next figure in the pattern shown below.

Find a counterexample for this statement:“The sum of two numbers is always greater than either addend.”

. . . 55 ,51 ,47

5) If a dart is thrown at the circle to the right, what is the probability that it will land in a shaded sector? The odds?

Points, Lines, and PlanesPoints, Lines, and Planes

Geometry is the study of points, lines, and planes and their relationships.

Everything we see contains elements of geometry.

Even the painting to the right is made entirely of small, carefully placed dots of color.

Georges Seurat, Sunday Afternoon on the Island of LeGrande Jatte, 1884 - 1886

You will learn to identify and draw models of points, lines, andplanes, and determine their characteristics.


A ____ is the basic unit of geometry.point

POINT: A point has no ____.size

Points are named using capital letters.

The points at the right are named point A and point B.

A

B


A ____is a series of points that extends without end in two directions.line

LINE: A line is made up of an ______ _______ of points.infinite number

The ______ show that the line extends without end in both directions.

A line can be named with a single lowercase script letter or by two points on the line.

arrows

The line below is named line AB, line BA, or line l. The symbol for line AB is AB

��

A

B

l


R

T

m S

1) Name two points on line m.

Possible answers:

point R and point S

point R and point T

point S and point T

2) Give three names for the line.

Possible answers:

or line m RS RT ST��

NOTE: Any two points on the line or the script letter can be used to name it.


Three points may lie on the same line. These points are _______ .collinear

Points that DO NOT lie on the same line are __________ .noncollinear

R

T

S

U

V

1) Name three points that are collinear.

Possible answers:

points R, S, and point T

points U, S, and point V


Three points may lie on the same line. These points are _______ .collinear

Points that DO NOT lie on the same line are __________ .noncollinear

R

T

S

U

V

1) Name three points that are noncollinear.

Possible answers:

points R, S, and point V

points R, S, and point U

points S, T, and point V points R, V, and point U

points R, T, and point V

points R, T, and point U


A ___ has a definite starting point and extends without end in one direction.ray

RAY: The starting point of a ray is called the ________.endpoint

A ray is named using the endpoint first, then another point on the ray. The ray above is named ray AB.

A B

Rays and line segments are parts of lines.

The symbol for ray AB is AB��


LINE SEGMENT:

A line segment is part of a line containing two endpoints and all points between them.

A line segment is named using its endpoints.

The line segment above is named segment AB or segment BA.

A B

Rays and line segments are parts of lines.

A ___________ has a definite beginning and end.line segment

The symbol for segment AB is AB


1) Name two segments.

A

C

B

U

DPossible Answers:

, and ,

, and ,

AB AC

BD BC

2) Name a ray.

Possible Answers:

, ,

,

AB AC

DB DU

��

��


A _____ is a flat surface that extends without end in all directions.plane

Points that lie in the same plane are ________.

Points that do not lie in the same plane are ___________.

coplanar

noncoplanar

PLANE:

For any three noncollinear points, there is only one plane that contains all three points.

A plane can be named with a single uppercase script letter or by three noncollinear points.

The plane at the right is named

plane ABC or plane M.

B

A M

C

Hands OnHands On

AAAA

BBBB

CCCC

DDDD

EEEE

Place points A, B, C, D, & E on a piece of paper as shown.

Fold the paper so that point A is on the crease.

Open the paper slightly. The two sections of the paper represent different planes.

1) Name three points that are coplanar. ______________________

2) Name three points that are noncoplanar. ______________________

3) Name a point that is in both planes. ______________________

Answers (may be others)

A, B, & C

D, A, & B

A


5-Minute Check5-Minute Check

D

F

r E

C

1) Name three points

on line r

2) Give three other names

for line r3) Name two segments that have point F as an endpoint.

4) Name three different rays.

5) Are points C, E, and F collinear or noncollinear?

D, E, F

, , DE DF EF��

, DF EF

, ( ), DC DF orDE EF��

noncollinear

PostulatesPostulates

You will learn to identify and use basic postulates about points, lines, and planes.


Geometry is built on statements called _________.

Postulates are statements in geometry that are accepted to be true.

postulates

Postulate 1-1: Two points determine a unique ___.

There is only one line that containsPoints P and Q

Q

P

Postulate 1-2: If two distinct lines intersect, then their intersection is a ____.

Lines l and m intersect at point T

Tl

m

Postulate 1-3: Three noncollinear points determine a unique _____.

There is only one plane that contains points A, B, and C.

A

BC

line

point

plane


A

BC

Points A, B, and C are noncollinear.

1) Name all of the different lines that can be drawn through these points.

AC��

CB��

BA��

2) Name the intersection of , and AC CB��

Point C


1) Name all of the planes that are represented in the figure.

There is only one plane that contains threenoncollinear points.

plane ABC (side)

B

A

D

C

plane ACD (side)

plane ABD (back side)

plane BCD (bottom)

M


N

D E

Postulate 1-4: If two distinct planes intersect, then their intersection is a ___.

Plane M and plane N intersect in line DE.

line


Name the intersection of plane CDG and plane BCD.

DC��

H

E F

G

CB

A D

Name two planes that intersect in .DF��

planes ADF and CDF



1) At which point or points do three planes intersect?

2) Name the intersection of plane ABC and plane ACD.

3) Are there two planes in the figure that do not intersect?

B

A

D

C

4) Name two planes that intersect in .BD��

5) How many points do and have in common?

AB��

BC��

At each of the points A, B, C, and D.

No

AC��

Planes ABD and BCD.

One, (Point B)

Conditional Statements and Their ConversesConditional Statements and Their Converses

In mathematics, you will come across many _______________.

For Example: If a number is even,

then it is divisible by two.

If – then statements join two statements based on a condition:

A number is divisible by two only if the number is even.

Therefore, if – then statements are also called __________ __________ .

if-then statements

conditional statements

You will learn to write statements in if-then form and writethe converse of the statements.


Conditional statements have two parts.

The part following if is the _________ .hypothesis

The part following then is the _________ .conclusion

If a number is even, then the number is divisible by two.a number is even the number is divisible by two.

Hypothesis:

Conclusion:


How do you determine whether a conditional statement is true or false?

ConditionalStatement

True orFalse

Why?

If it is the 4th of July(in the U.S.), then it is aholiday.

True The statement is true becausethe conclusion follows fromthe hypothesis.

If an animal lives in the water, then it is a fish.

False You can show that the statement is false by givingone counterexample.

Whales live in water, butwhales are mammals, not fish.


There are different ways to express a conditional statement.The following statements all have the same meaning.

If you are a member of Congress, then you are a U.S. citizen.

All members of Congress are U.S. citizens.

You are a U.S. citizen if you are a member of Congress.

You write two other forms of this statement: “If two lines are parallel, then they never intersect.”

All parallel lines never intersect.

Lines never intersect if they are parallel.

Possible answers:


The ________ of a conditional statement is formed by exchanging the hypothesis and the conclusion.

converse

Conditional: If a figure is a triangle, then it has three angles.a figure is a triangle it has three angles

Converse: If _______________, then ________________.

NOTE: You often have to change the wording slightly so that the converse reads smoothly.

Converse: If the figure has three angles, then it is a triangle.


Write the converse of the following statements.State whether the converse is TRUE or FALSE.If FALSE, give a counterexample:

“If you are at least 16 years old, then you can get a driver’s license.”

If ________________________, then _______________________.you can get a driver’s license you are at least 16 years old

“If today is Saturday, then there is no school.

If _______________, then ______________.there is no school today is Saturday

TRUE!

FALSE!

We don’t have school on New Years day which may fall on a Monday.



If the power goes out, we will light candles.

1) Identify the hypothesis and conclusion of the statement.

2) Write two other forms of the statement.

3) Write the converse of the statement.

4) Is the converse you wrote for # 3 (above) true?

Hypothesis: the power goes outConclusion: we will light candles

1) We will light candles if the power goes out.2) Whenever the power goes out, we will light candles

If we light candles, then the power has gone out.

NO! You could light candles for another reason, such as a birthday party.

Tools of the TradeTools of the Trade

You will learn to use geometry tools.


As you study geometry, you will use some of the basic tools.

A __________ is an object used to draw a straight line.

A credit card, a piece of cardboard, or a ruler can serve as a straightedge.

straightedge

Determine whether thesides of the triangle arestraight.

Place a straightedgealong each side of thetriangle.


A _______ is another useful tool. compass

A common use for a compass is drawing arcs and circles.(an arc is part of a circle)


D

CBA

Use a compass to determine which segment is longer or AC BD

1) Place the point of the compass on A and adjust the compass so that the pencil is on C.

2) Without changing the setting of the compass, place the point of the compass on B. The pencil point does not reach point D. Therefore, is longer.BD

These drawings are called ___________ . constructions

In geometry, you will draw figures using only a compass and a straightedge.


Use a compass and straightedge to construct a six-sided figure.

1) Use the compass draw a circle.2) Using the same compass setting, put the point on the circle and draw a small arc on the circle.3) Move the compass point to the arc and draw another arc along the circle. Continue doing this until there are six arcs.4) Use a straightedge to connect the points in order.

Constructing the MidpointConstructing the Midpoint

You will learn to construct the midpoint of a line segment using only astraightedge and compass.

1) On your patty paper, draw two points.

2) Construct a line segment between the points

3) Fold the paper, and place one point on top of the other. This should produce a crease (fold mark) between the points.

4) Place the compass on one of the points and open it to over half way to the other point.

5) Repeat step 4 using the second point.

6) Connect the intersection of the two circles.


A Plan for Problem SolvingA Plan for Problem Solving

You will learn to solve problems that involve the perimetersand areas of rectangles and parallelograms.

Perimeter is the _____________________.distance around an object

Perimeter is similar to ____________.a line segment

Area is the _______________________________________________.number of square units needed to cover an object’s surface

Area is similar to ______.a plane


In this section you will learn to solve problems that involve the perimetersand areas of rectangles and parallelograms.

Perimeter is the ____________________.distance around a figure

The perimeter is the ____ of the lengths of the sides of the figure.sum

The perimeter of the room shown here is:

15 ft + 12 ft+ 18 ft + 6 ft + 6 ft + 9 ft

= 66 ft


Some figures have special characteristics. For example, the opposite sidesof a rectangle have the same length.

This allows us to use a formula to find the perimeter of a rectangle.(A formula is an equation that shows how certain quantities are related.)

Perimeter 2 2

2( )

l w

l w

(of a rectangle)


Find the perimeter of a rectangle with a length of 17 ft and a width of 8 ft.

Perimeter 2 2 or 2( )l w l w (of a rectangle)

17 ft

8 ft

= 2(17 ft) + 2(8 ft) = 2(17 ft + 8 ft)

= 34 ft + 16 ft

= 50 ft

= 2(25 ft)

= 50 ft


Another important measure is area.

The area of a figure is ____________________________________________.the number of square units needed to cover its surface

The area of the rectangle below can be found by dividing it into 18 unit squares.

3

6

The area of a rectangle can also be found by multiplying the length and the width.


The area “A” of a rectangle is the product of the length l and the width w.

l

w A lw

Find the area of the rectangle

14 in.

10 in.A lw

(14 )(10 )A in in2140A in

The area of the rectangle is 140 square inches.

NOTE: units indicate area is being calculated 2( )( )in in in

Plan for Problem SolvingPlan for Problem Solving

Because the opposite sides of a parallelogram have the same length,the area of a parallelogram is closely related to the area of a ________.rectangle

The area of a parallelogram is found by multiplying the ____ and the ______.base height

base

height

Base – the bottom of a geometric figure.Height – measured from top to bottom, perpendicular to the base.


Find the area of the parallelogram:

4 m

15

10m

4.3 m

51 (4 )

10

A bh

m m

2

2

204

10

220

5

m

m

§1.6 A Plan for Problem Solving§1.6 A Plan for Problem Solving

§1.5 Tools of the Trade§1.5 Tools of the Trade

Documents

§ 1.1 Patterns and Inductive Reasoning Patterns and Inductive ReasoningPatterns and Inductive Reasoning § 1.4 Conditional Statements and Their Converses