Transcript conditional statement

Chapter 2
Reasoning and Proof
Chapter Objectives
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
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Recognize conditional statements
Compare bi-conditional statements and
definitions
Utilize deductive reasoning
Apply certain properties of algebra to
geometrical properties
Write postulates about the basic components of
geometry
Derive Vertical Angles Theorem
Prove Linear Pair Postulate
Identify reflexive, symmetric and transitive
Lesson 2.1
Conditional Statements
Lesson 2.1 Objectives
Analyze conditional statements
 Write postulates about points, lines,
and planes using conditional
statements
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Conditional Statements
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A conditional statement is any
statement that is written, or can be
written, in the if-then form.

This is a logical statement that contains
two parts
• Hypothesis
• Conclusion
If today is Tuesday, then tomorrow is Wednesday.
Hypothesis

The hypothesis of a conditional
statement is the portion that has, or
can be written, with the word if in
front.

When asked to identify the hypothesis,
you do not include the word if.
If today is Tuesday, then tomorrow is Wednesday.
Conclusion

The conclusion of a conditional
statement is the portion that has, or
can be written with, the phrase
then in front of it.

Again, do not include the word then
when asked to identify the conclusion.
If today is Tuesday, then tomorrow is Wednesday.
Converse

The converse of a conditional
statement is formed by switching
the hypothesis and conclusion.
If today is Tuesday, then tomorrow is Wednesday.
If tomorrow is Wednesday, then today is Tuesday
Negation

The negation is the opposite of the
original statement.
Make the statement negative of what it
was.
 Use phrases like

• Not, no, un, never, can’t, will not, nor,
wouldn’t, etc.
Today is Tuesday.
Today is not Tuesday.
Inverse
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The inverse is found by negating
the hypothesis and the conclusion.

Notice the order remains the same!
If today is Tuesday, then tomorrow is Wednesday.
If today is not Tuesday, then tomorrow is not
Wednesday.
Contrapositive

The contrapositive is formed by
switching the order and making both
negative.
If today is Tuesday, then tomorrow is Wednesday.
If today is not Tuesday, then tomorrow is not
Wednesday.
If tomorrow is not Wednesday, then today is not
Tuesday.
Point, Line, Plane Postulates:
Postulate 5
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Through any two points there exists
exactly one line.
Y
O
Point, Line, Plane Postulates:
Postulate 6
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A line contains at least two points.

Taking Postulate 5 and Postulate 6 together
tells you that all you need is two points to
make one line.
H
I
Point, Line, Plane Postulates:
Postulate 7
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If two lines intersect, then their
intersection is exactly one point.
B
Point, Line, Plane Postulates:
Postulate 8
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Through any three noncollinear
points there exists exactly one
plane.
R
M
L
Point, Line, Plane Postulates:
Postulate 9
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A plane contains at least three
noncollinear points.
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Take Postulate 8 with Postulate 9 and this
says you only need three points to make a
plane.
R
M
L
Point, Line, Plane Postulates:
Postulate 10
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If two points lie in a plane, then the
line containing them lies in the
same plane.
M
E
Point, Line, Plane Postulates:
Postulate 11
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If two planes intersect, then their
intersection is a line.

Imagine that the walls of the classroom are
different planes.
• Ask yourself where do they intersect?
• And what geometric figure do they form?
Homework 2.1
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In Class

1-8
• p75-78
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Homework
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10-50 ev, 51, 55, 56
Due Tomorrow
Lesson 2.2
Definitions
and
Biconditional
Statements
Lesson 2.2 Objectives
Recognize a definition
 Recognize a biconditional statement
 Verify definitions using biconditional
statements
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Perpendicular Lines
Perpendicular lines intersect to
form a right angle.
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When writing that lines are
perpendicular, we place a special
symbol between the line segments
• AB
T
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CD
Definition
The previous slide was an example
of a definition.
 It can be read forwards or
backwards and maintain truth.
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Biconditional Statement
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A biconditional statement is a
statement that is written, or can be
written, with the phrase if and only if.
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If and only if can be written shorthand by iff.
Writing a biconditional is equivalent to
writing a conditional and its converse.
All definitions are biconditional
statements.
Finding Counterexamples
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To find a counterexample, use the
following method
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Assume that the hypothesis is TRUE.
Find any example that would make the
conclusion FALSE.
For a biconditional statement, you must
prove that both the original conditional
statement has no counterexamples and
that its converse has no
counterexamples.

If either of them have a counterexample, then
the whole thing is FALSE.
Example 1
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If a+b is even, then both a and b
must be even.
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Assume that the hypothesis is TRUE.
• So pick a number that is even (larger than 2)
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Find any example that would make the
conclusion FALSE.
• Pick two numbers that are not even but add to
equal the even number from above.
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Those two numbers you picked are your
counterexample.
If no counterexample can be found, then the
statement is true.
Homework 2.2
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In Class
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3-12
• p82-85
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Homework
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14-42 even
Due Tomorrow
Lesson 2.3
Deductive Reasoning
Lesson 2.3 Objectives
Use symbolic notation to represent
conditional statements
 Identify the symbol for negation
 Utilize the Law of Detachment to
form conclusions
 Utilize the Law of Syllogism to form
conclusions
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Symbolic Conditional Statements
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To represent the hypothesis symbolically,
we use the letter p.
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We are applying algebra to logic by
representing entire phrases using the letter p.
To represent the conclusion, we use the
letter q.
To represent the phrase if…then, we use
an arrow, .
To represent the phrase if and only if, we
use a two headed arrow,
.
Example of Symbolic Representation
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If today is Tuesday, then tomorrow is
Wednesday.
p=
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q=
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Today is Tuesday
Tomorrow is Wednesday
Symbolic form
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pq
• We read it to say “If p then q.”
Negation
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Recall that negation makes the
statement “negative.”
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That is done by inserting the words not,
nor, or, neither, etc.
The symbol is much like a negative
sign but slightly altered…

~
Symbolic Variations
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Converse
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Inverse
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~p  ~q
Contrapositive
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qp
~q  ~p
Biconditional
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p
q
Logical Argument
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Deductive reasoning uses facts, definitions, and
accepted properties in a logical order to write a
logical argument.
So deductive reasoning either states laws and/or
conditional statements that can be written in
if…then form.
There are two laws that govern deductive
reasoning.
If the logical argument follows one of those
laws, then it is said to be valid, or true.
Law of Detachment
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If pq is a true conditional statement
and p is true, then q is true.
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It should be stated to you that pq is true.
Then it will describe that p happened.
So you can assume that q is going to happen
also.
This law is best recognized when you are
told that the hypothesis of the conditional
statement happened.
Example 2
If you get a D- or above in
Geometry, then you will get credit
for the class.
 Your final grade is a D.
 Therefore…
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You will get credit for this class!
Law of Syllogism
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If pq and qr are true conditional
statements, then pr is true.
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This is like combining two conditional
statements into one conditional statement.
• The new conditional statement is found by taking
the hypothesis of the first conditional and using the
conclusion of the second.
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This law is best recognized when multiple
conditional statements are given to you
and they share alike phrases.
Example 3
If tomorrow is Wednesday, then the
day after is Thursday.
 If the day after is Thursday, then
there is a quiz on Thursday.
 Therefore…
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And this gets phrased using another
conditional statement
• If tomorrow is Wednesday, then there is a
quiz on Thursday.
Deductive v Inductive Reasoning
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Deductive reasoning
uses facts, definitions, and
accepted properties in a
logical order to write a
proof.
This is often called a
logical argument.
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Inductive reasoning
uses patterns of a sample
population to predict the
behavior of the entire
population
This involves making
conjectures based on
observations of the sample
population to describe the
entire population.
If the conditional statement is true, then the contrapositive
is also true. Therefore they are equivalent statements!
Equivalent Statements
Conditional Converse
Inverse Contrapositive
If p, then q
If q, then p
If ~p,
then ~q
If ~q, then ~p
Written just as
it shows in the
problem.
Switch the
hypothesis
with the
conclusion.
Take the
original
conditional
statement
and make
both parts
negative.
Take the converse
and make both
parts negative.
If the converse is true, then the inverse is also true.
Therefore they are equivalent statements!
Means “not”
Homework
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In Class
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1-5
• p91-94
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Homework
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8-48 even
Due Tomorrow
Lesson 2.4
Reasoning with
Properties of
Algebra
Lesson 2.4 Objectives
Use properties from algebra to
create a proof
 Utilize properties of length and
measure to justify segment and
angle relationships

Algebraic Properties of Equality
Property
Addition
Property
Definition
Identification
Something is added to both
If a=b, then a+c = b+c. sides of the equation.
Abbreviation
APOE
If a=b, then a-c = b-c.
Something is subtracted
from both sides of the
equation.
SPOE
Multiplication
Property
If a=b, then ac = bc.
Something is multiplied to
both sides of the equation.
MPOE
Division
Property
If a=b and c≠0, then
a/c = b/c.
Something is being divided
into both sides.
DPOE
Substitution
Property
If a=b, then a can be
substituted for b in any
expression.
One object is used in place
of another without any
calculations being done.
SUB
a(b+c) = ab + ac
A number outside of
parentheses has been
multiplied to all numbers
inside.
DIST
Subtraction
Property
Distributive
Property
Reflexive, Symmetric and Transitive
Properties
Definition
How
to
Remember
How to
Use
Reflexive
Symmetric
Transitive
For any real
number a,
a=a
If a=b, then
b=a.
If a=b and b=c,
then a=c.
Reflexive is close to
reflection, which is
what you see when you
look in a mirror.
Symmetric starts
with s, so that
means to switch
the order.
Transitive is like
transition, and when
a and c equal the
same thing, they
must transition to
equal each other.
This will be used when two
objects share something,
such as sharing a common
side of a triangle
This is a step that
allows you to
change the order of
objects so they fit
where you need
them.
This is used most often
in proofs, and can be
often thought of as
substitution.
Show Your Work
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This section is an introduction to proofs.
To solve any algebra problem, you now
need to show ALL steps.
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And with those steps you need to give a
reason, or law, that allows you to make that
step.
Remember to list your first step by simply
rewriting the problem.

This is to signify how the problem started.
Example 4
Solve 9x+18=72
9x+18=72
Given
9x=54
SPOE
-18
9
-18
9
x=6
DPOE
Short for “Information
given to us.”
Example 5: Using Segments
In the diagram, AB=CD. Show that AC=BD.
A
C
B
AB=CD
Given
AB+BC=BC+CD
APOE
Think about changing AB into
AC? And the same with CD into
BD?
AC=AB+BC
Segment Addition
Postulate
BD=BC+CD
Segment Addition
Postulate
AC=BD
D
Transitive POE
Example 6: Using Angles
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HW Problem #24, p100
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In the diagram, m RPQ=m RPS, verify to
show that m SPQ=2(m RPQ).
mRPQ=m RPS
m SPQ=m RPQ+m RPS
m SPQ=m RPQ+m RPQ
m SPQ=2(m RPQ)
Given
Angle Addition
Postulate
S
SUB
R
DIST
P
Q
Example 7
Fill in the two-column proof with the appropriate reasons for each step
APOE
MPOE
Symmetric POE
Homework 2.4
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In Class
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1,4-8
• p99-101
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Homework
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10-32, 36-50 even
Due Tomorrow
Lesson 2.5
Proving Statements
about Segments
Lesson 2.5 Objectives
Write a two-column proof
 Justify statements about congruent
segments

Theorem

A theorem is a true statement that
follows the truth of other
statements.


Theorems are derived from postulates,
definitions, and other theorems.
All theorems must be proved.
Two-Column Proof

One method of proving a theorem is to use a
two-column proof.

A two-column proof has numbered statements and
corresponding reasons placed in a logical order.
• That logical order is just steps to follow much like reading
a cook book.

The first step in a two-column proof should
always be rewriting the information given to you
in the problem.

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When you write your reason for this step, you say
“Given”.
The last step in a two-column proof is the exact
statement that you are asked to show.
Example 8
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Prove the Symmetric Property of Segment
Congruence.
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GIVEN: Segment PQ is congruent to Segment XY
PROVE: Segment XY is congruent to Segment PQ
Hints for Making Proofs
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Remember to always write down the first step as
given information.
Develop a mental plan of how you want to
change the first statement to look like the last
statement.
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Try to evaluate how you can make each step change
from the previous by applying some rule.
You must follow the postulates, definitions, and
theorems that you already know.
Number your steps so the statements and the
reasons match up!
Example 9
Fill in the missing steps
Transitive POE
A  C
Example 10
Fill in the missing steps
1 and 2 are a linear pair
1 and 2 are supplementary
Definition of supplementary angles
m1 = 180o - m2
Homework 2.5

In Class

1,3-5,7,9
• p105-107

Homework


6-11,16,21,22
Due Tomorrow
Lesson 2.6
Proving Statements
about Angles
Lesson 2.6 Objectives
Utilize the angle and segment
congruence properties
 Prove properties about special angle
pairs
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Theorem 2.1:
Properties of Segment Congruence
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Segment congruence is always

Reflexive
• Segment AB is congruent to Segment AB.
Symmetric
• If AB  CD, then CD  AB.
 Transitive
• If AB  CD and CD  EF, then AB  EF.
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Theorem 2.2:
Properties of Angle Congruence
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Angle congruence is always
Reflexive
• A  A
 Symmetric
• If A  B, then B  A.
 Transititve
• If A  B and B  C, then A  C.

Theorem 2.3:
Right Angle Congruence Theorem

All right angles are congruent.
GIVEN: 1 and 2 are right angles.
PROVE: 1  2
1
2
1. 1 and 2 are right angles
1. Given
2. m1 = 90o, m2 = 90o
2. Definition of Right Angles
3. m1 = m2
3. Trans POE
4. 1  2
4. DEFCON
Theorem 2.4:
Congruent Supplements Theorem

If two angles are supplementary to
the same angle, or congruent
angles, then they are congruent.

If m1 + 2 = 180o and m2 + m3 = 180o,
then 1  3.
1
2
3
Theorem 2.5:
Congruent Complements Theorem

If two angles are complementary to
the same angle, or to congruent
angles, then they are congruent.

If m4 + m5 = 90o and m5 + m6 =90o,
then 4  6.
4
5
6
Postulate 12:
Linear Pair Postulate

The Linear Pair Postulate says if two
angles form a linear pair, then they are
supplementary.
1 + 2 = 180o
1
2
Theorem 2.6:
Vertical Angles Theorem


If two angles are vertical angles,
then they are congruent.
Vertical angles are angles formed by the intersection of
two straight lines.
1  3
1
2
4
3
2  4
Example 11
Using the following figure, fill in the
missing steps to the proof.
Given
2
Definition of a linear pair
4
m1 + m2 = 180o
m3 + m4 = 180o
Congruent Supplements Theorem
Homework 2.6

In Class

1,3-9,10,23
• p112-116

Homework


10, 12-22, 27-28, 33-36
Due Tomorrow