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)
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, …
(12.)
Examples…Draw the next figure in the pattern.
(11.)
(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
2 and 8
Product
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.
Questions to think about:
•
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.
DESCRIPTION
NAMING
CONVENTIONS
DIAGRAM
LINE
POINT
TERM
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DESCRIPTION
NAMING
CONVENTIONS
DIAGRAM
OPPOSITE RAYS
RAY
LINE SEGMENT
PLANE
TERM
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DESCRIPTION
NAMING
CONVENTIONS
DIAGRAM
COPLANAR
(NONCOPLANAR)
COLLINEAR
(NONCOLLINEAR)
TERM
Examples…Given the following diagram answer the questions.
(24.)
•
C
m
Name two points on line m.
•
•
•
P
Q
R
•
(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.
A
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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
1.1
WORDS
MODELS
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
Three noncollinear points determine
a unique plane.
1.3
•B
•A
There is only one plane that
contains points A, B, and C.
•C
If two distinct planes intersect, then
their intersection is a line.
1.4
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.
Example….Using the diagram to the
right answer the question.
(38.)
Using your picture in the prior problem. Name the
intersection of DE and EF .
(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
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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.
Questions to think about:
•
How can you conclude that a conditional statement is true?
Definition
A conditional statement is an if…then…statement.
Characteristics
p q
If p then q
P implies q
If this happens, then that happens.
The hypothesis is the part following the if.
The conclusion is the part following then.
conditional
statement
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.
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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
converse
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.
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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.
Questions to think about:
•
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?
(64.)
Which of the following claims would be best
expressed by deductive reasoning?
a.
Your first quiz grade usually indicates how
you will do in the course.
a.
California's population growth rate slowed
last year.
b.
The final exam accounts for 30% of the
course grade.
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.
c.
d.
Late papers will not be accepted.
Gravity's Rainbow is required reading in
your course.
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(65.)
Inductive or deductive?
(66.)
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.
(67.)
Inductive or deductive?
Inductive or deductive?
Carla has had a quiz in science every Friday for the
last two weeks. She concludes that she will have a
quiz this Friday.
(68.)
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.
Inductive or deductive?
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.
(70.)
(2) Amanda is older than Teresa.
(2) Luke is 16 today.
(71.)
(1) If x is an integer, then x is a real number.
(2) x is an integer
(1) If Amanda if taller than Teresa, then Amanda is
at least 6 feet tall.
(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.
(74.)
(2) 5 and 3 are added.
(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.
(1) If the measure of an angle is less than 90, it is
an acute angle.
(2) m∠B = 45
(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.
(78.)
(1) If I work part, I will save money.
(2) If I save money, I can buy a computer.
(2)If the sum of the measures of two angles is 90,
then the angles are complementary.
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(79.)
(1) If two angles are vertical angles, then they are
congruent.
(80.)
(1).if it is sunny tomorrow, I will go swimming
(2) If I go swimming, I will miss baseball.
(2) If two angles are congruent, then their
supplements are congruent.
(81.)
If a parallelogram has four congruent sides, it is a
rhombus. If a figure is a rhombus, then the
diagonals are perpendicular.
(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.
(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 *)
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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