Chapter 1 Reasoning in Geometry - East Penn School...

Geometry Concepts

Chapter 1 Reasoning in Geometry 1.1 Identify patterns (2.1)

1.1 Use inductive reasoning (2.1)

1.2 Identify models of points, lines, and planes (1.2)

1.3 Identify and use postulates about points, line, and planes (1.2)

1.4 Write statements in if-then form (2.2)

1.4 Write statements in converse form (2.2)

15.2 Use deductive reasoning (2.4)

15.2 Use the law of detachment (2.4)

15.2 Use the law of syllogism (2.4)

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Section 1.1 Patterns and Inductive Reasoning

There are two types of reasoning inductive and deductive reasoning.

Inductive reasoning is based on patterns, observations and/or past events.

Deductive reasoning is based on facts, rules, laws, and/or definitions.

Questions to think about:

• Can you name any examples of number patterns?

• Can you explain how recognizing number patterns can help when making a decision?

• Can you think of a game that uses patterns?

• How do you use inductive reasoning to make a conjecture?

Examples…Find the next three terms of each sequence.

(1.) 33, 39, 45, … (2.) 1.25, 1.45, 1.65, …

(3.) 1, 3, 9, … (4.) 13, 8, 3, …

(5.) 32, 16, 8, … (6.) 6, 12, 24, …

Examples…Find the next three terms of each sequence.

(7.) 1, 3, 7, 13, 21, … (8.) 10, 12, 15, 19, …

(9.) 1, 2, 6, 24, … (10.) 101, 102, 105, 110, 117, …

Examples…Draw the next figure in the pattern.

(11.) �� (12.) ��

(13.) (14.)

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A conjecture is a conclusion that we reach based on inductive reasoning. This is an

educated guess. Sometimes it will be true or it could be false. If you find one example that

does not comply with your conjecture then you have discovered a counterexample (similar

to our nonexamples). It is important to gather enough data before you make a conjecture.

Examples…

(15.) Look at the circles. What conjecture can you make about the number of regions 20 diameters forms?

(16.) Justin studied the data in the table and made the

following conjecture. “The product of two positive

numbers is always greater than either factor”. Find

the counterexample for his conjecture.

Factors Product

2 and 8 16

5 and 15 75

20 and 38 760

54 and 62 3348

(17.) Aaron studied the data below and made the

following conjecture. “Multiplying a number by -1

produces a product that is less than -1” Find a

counterexample for his conjecture.

5(-1) = -5 and -5 < -1

15(-1) = -15 and -15 < -1

100(-1) = -100 and -100 < -1

300(-1) = -300 and -300 < -1

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(18.) Find a counterexample. If a flower is red, it is a

rose.

(19.) Find a counterexample. When you multiply a

number by 2, the product is greater than the

original number.

(20.) What is a counterexample for the conjecture “when

you multiply a number by 3, the product is divisible

by 6.”

(21.) Predict the next term in the sequence.

12345679 x 9 = 111111111

12345679 x 18 = 222222222

12345679 x 27 = 333333333

12345679 x 36 = 444444444

12345679 x 45 = ___________

(22.) Write a conjecture…the sum of two odd numbers is

___________”

(23.) Write a conjecture. The product of two even

numbers is ____________.

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Section 1.2 Points, Lines and Planes

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

see contains elements of geometry. When looking at these definitions take note that the

word point is used in the description of a line, and the word line is used in the description

of plane.

A point is the basic unit of geometry. A line is made up of an infinite number of points.


• Can you explain the difference between a line and a segment?

• How is naming a ray similar to naming a line? How is it different?

• Why do you use two arrowheads when drawing or naming a line such as RS ?

• Can three points shown on a line be used to name a plane? Explain.

TERM DESCRIPTION NAMING CONVENTIONS

DIAGRAM

POINT

LINE

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PLANE

LINE SEGMENT

RAY

OPPOSITE RAYS


DIAGRAM

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• P

• Q

• R

• A

• C

m

COLLINEAR

(NONCOLLINEAR)

COPLANAR

(NONCOPLANAR)

Examples…Given the following diagram answer the questions.

(24.) Name two points on line m.

(25.) Give three names for the line.

(26.) Name two line segments. (27.) Name two rays.

(28.) Name three collinear points. (29.) Name three noncollinear points.


DIAGRAM

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Examples…Given the following diagram answer the questions.

(30.) What are two ways to name the line QT?

(31.) What are two other ways to name plane P?

(32.) What are the names of three different collinear points?

(33.) What are the names of four coplanar points?

(34.) What are two points that are not coplanar with points R, S, and V?

(35.) Why is RQS not a valid name for the plane?

(36.) What point(s) do lines l and m have in common?

Section 1.3 Postulates

Postulates are statements in geometry that are accepted as true.

POSTULATE WORDS MODELS

1.1 Through any two points there is exactly one line.

There is only one line contains A and B.

1.2 If two distinct lines intersect, then their intersection is exactly one point.

Lines CB and GH intersect at I.

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POSTULATE WORDS MODELS

1.3 Three noncollinear points determine a unique plane.

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

1.4 If two distinct planes intersect, then their intersection is a line.

Plane M and plane C intersect in line AP.

Examples…

(37.) Points D, E, and F are noncollinear. Draw the

diagram. Name all the different lines that can be

drawn through these points.

(38.) Using your picture in the prior problem. Name the

intersection of DE and EF .

Example….Using the diagram to the

right answer the question.

(39.) Name all the planes.

Example….Using the diagram to the right answer the question.

(40.) Name the intersection of plane CDG and plane BCD. (41.) Name two planes that intersect

• A • B

• C

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Section 1.4 Conditional Statements and their converses

In math and real life you will hear many if-then statements. For example, if you eat your

dinner than you will get snack or if you do your chores then you will receive your allowance

or if a number is even then it will be divisible by 2. These statements are considered

conditional because both statements are based on conditions. Take note that the second

part of the statement occurs only if the first part of has occurred. Think about this…

This is a true bumper sticker……..If you can read this, then you are too close.

Are these other bumper stickers true or false? Explain.

1. If you are too close, then you can read this.

2. If you cannot read this, then you are not too close.

3. If you are not too close, then you cannot read this.


• How can you conclude that a conditional statement is true?

Definition

A conditional statement is an if…then…statement.

The hypothesis is the part following the if.

The conclusion is the part following then.

Characteristics

p �q

If p then q

P implies q

If this happens, then that happens.

Examples Nonexamples

Examples….Identify the hypothesis and conclusion in the statements.

(42.) If it is Saturday, then Aaron plays soccer.

(43.) If two lines intersect, then their intersection is a point.

conditional

statement

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(44.) If you are an American citizen, then you have the right to vote.

(45.) If you want to be healthy, then you should eat vegetables.

(46.) If an animal is a robin, then the animal is a bird.

(47.) If an angle measures 130, then the angle is obtuse.

(48.) School will be cancelled if it snows more than six inches.

Examples….Writing a Conditional (step one: identify the hypothesis and conclusion; step two: write the

conditional)

(49.) All spiders are insects. (50.) Vertical angles share a vertex.

(51.) When it rains I use an umbrella. (52.) 3x – 7 = 14 implies that 3x = 21.

(53.) You will win the race if you swim faster. (54.) There is no school on Sunday.

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Definition

The converse of a conditional statement is formed by

exchanging the hypothesis and conclusion.

Characteristics

q �p

If q then p

q implies p

not always true

may have to change wording

Examples Nonexamples

Examples….Write the converse of this statement. Is the converse statement true?

(55.) If a figure is a triangle, then it has three angles. (56.) If you are at least 16 years old, then you can get a

drivers license.

(57.) If you are a quarterback, then you play football. (58.) Pianists are musicians.

(59.) All numbers divisible by five end in zero or five. (60.) Loud music will cause hearing loss.

(61.) If today is Saturday, then there is no school. (62.) If the geometry test is a 93%, then you will get an

A.

converse

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Section 15.2 Deductive Reasoning

Deductive reasoning (sometimes called logical reasoning) is based on facts, rules, laws,

and/or definitions. You use deductive reasoning when you solve a puzzle or play a game.

Justin: I've noticed previously that every time I kick a ball up, it comes back down, so I

guess this next time when I kick it up, it will come back down, too.

Aaron: That's Newton's Law. Everything that goes up must come down. And so, if you kick

the ball up, it must come down.

Justin’s reasoning is inductive reasoning because he has made a statement based on a

pattern that was observed. Aaron’s reasoning is deductive because the statement is based

on a law.


• Explain the difference between inductive and deductive reasoning.

• How is deductive reasoning different from inductive reasoning?

• If only one of the laws is to be used to reach a conclusion, how can you quickly determine which to use?

Examples….

(63.) Which of the following claims would be best

expressed by inductive reasoning?

a. Your first quiz grade usually indicates how

you will do in the course.

b. The final exam accounts for 30% of the

course grade.

c. Late papers will not be accepted.

d. Gravity's Rainbow is required reading in

your course.

(64.) Which of the following claims would be best

expressed by deductive reasoning?

a. California's population growth rate slowed

last year.

b. California residents appreciate their good

weather.

c. California residents are residents of the

United States.

d. More cars are registered in California than

in any other state.

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(65.) Inductive or deductive?

Josh’s little sister found a nest of strange eggs near

the beach. The first five eggs hatched into lizards.

She concluded that all of the eggs were lizard eggs.


Carla has had a quiz in science every Friday for the

last two weeks. She concludes that she will have a

quiz this Friday.


Vinnie’s geometry teacher told his classes at the

beginning of the year that there would be a quiz

every Friday. Vinnie concludes that he will have a

quiz this Friday.


A number is divisible by 4 if its last two digits make

a number that is divisible by 4. Dean concluded

that 624 is divisible by 4.

Law of Detachment

If the hypothesis of a true conditional statement is true, then the conclusion is true.

If p � q is true and p is true, then q is true.

Examples….Use the Law of Detachment to determine a conclusion that follows from statements (1) and (2). If a

valid conclusion does not follow, write no valid conclusion.

(69.) (1) If you are at least 16 years old, then you can get

a driver’s license.

(2) Luke is 16 today.

(70.) (1) If Amanda if taller than Teresa, then Amanda is

at least 6 feet tall.

(2) Amanda is older than Teresa.

(71.) (1) If x is an integer, then x is a real number.

(2) x is an integer

(72.) (1) If a figure is a square, it has four right angles.

(2) A figure has four right angles.

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(73.) (1) If two odd numbers are added, their sum is an

even number.

(2) 5 and 3 are added.

(74.) (1) If the measure of an angle is less than 90, it is

an acute angle.

(2) m∠B = 45

(75.) If a student gets an A on a final exam, then the

students will pass the course. Felicia got an A on her

history final exam.

(76.) If two angles are adjacent, then they share a

common vertex. ∠1 and ∠2 share a common

vertex.

Law of Syllogism (squeeze play)

If p � q is true and q � ris true, then p � r is true.

If it is July then you are on summer vacation.

If you are on summer vacation, then you work at a smoothie shop.

You conclude: If it is July then you work at a smoothie shop.

Examples….Use the Law of Syllogism to determine a conclusion that follows from statements (1) and (2). If a

valid conclusion does not follow, write no valid conclusion.

(77.) (1)If a triangle is a right triangle, the sum of the

measures of the acute angle is 90.

(2)If the sum of the measures of two angles is 90,

then the angles are complementary.

(78.) (1) If I work part, I will save money.

(2) If I save money, I can buy a computer.

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(79.) (1) If two angles are vertical angles, then they are

congruent.

(2) If two angles are congruent, then their

supplements are congruent.

(80.) (1).if it is sunny tomorrow, I will go swimming

(2) If I go swimming, I will miss baseball.

(81.) If a parallelogram has four congruent sides, it is a

rhombus. If a figure is a rhombus, then the

diagonals are perpendicular.

(82.) If you improve your vocabulary,, then you will

improve your score on a standardized test. If you

read often, then you will improve your vocabulary.

(* you can switch the sentences *)

(83.) The following statements are true.

a. If Maria is drinking juice, then it is

breakfast.

b. If it is lunchtime, then Kira is drinking milk

and nothing else.

c. If it is a mealtime, then Curtis is drinking

water and nothing else.

d. If it is breakfast time, then Julio is drinking

juice and nothing else.

e. Maria is drinking juice.

Use the information given at the left. For each statement,

write must be true, may be true, or is not true. Explain

your reasoning.

(84.) Julio is drinking juice

(85.) Curtis is drinking water

(86.) Kira is drinking milk

(87.) Curtis is drinking juice

(88.) Maria is drinking water

(89.) Julio is drinking milk

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