Upload
noah-blake
View
238
Download
2
Tags:
Embed Size (px)
Citation preview
Chapter 2
Reasoning and Proof
Chapter Objectives 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
Conditional Statements 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 tomorrow is Wednesday,
If today is Tuesday, then 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 The inverse is found by negating
the hypothesis and the conclusion. Notice the order remains the same!
If today is not Tuesday,
If today is Tuesday, then tomorrow is Wednesday.
then tomorrow is not Wednesday.
Contrapositive The contrapositive is formed by
switching the order and making both negative.
If tomorrow is not Wednesday,
If today is Tuesday, then tomorrow is Wednesday.
If today is not Tuesday, then tomorrow is not Wednesday.
then today is not Tuesday.
Point, Line, Plane Postulates:Postulate 5
Through any two points there exists exactly one line.
Y O
Point, Line, Plane Postulates:Postulate 6 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
If two lines intersect, then their intersection is exactly one point.
B
Point, Line, Plane Postulates:Postulate 8
Through any three noncollinear points there exists exactly one plane.
M
R
L
Point, Line, Plane Postulates:Postulate 9 A plane contains at least three
noncollinear points. Take Postulate 8 with Postulate 9 and this
says you only need three points to make a plane.
M
R
L
Point, Line, Plane Postulates:Postulate 10
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 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 In Class
1-8• p75-78
Homework 10-50 ev, 51, 55, 56
Due Tomorrow
Lesson 2.2
DefinitionsandBiconditional Statements
Lesson 2.2 Objectives Recognize a definition Recognize a biconditional statement Verify definitions using biconditional
statements
Perpendicular Lines Perpendicular lines intersect to
form a right angle. When writing that lines are
perpendicular, we place a special symbol between the line segments• AB CD
T
Definition The previous slide was an example
of a definition. It can be read forwards or
backwards and maintain truth.
Biconditional Statement A biconditional statement is a
statement that is written, or can be written, with the phrase if and only if. 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 To find a counterexample, use the following
method 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 If a+b is even, then both a and b
must be even. Assume that the hypothesis is TRUE.
• So pick a number that is even (larger than 2) 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. Those two numbers you picked are your
counterexample. If no counterexample can be found, then the
statement is true.
Homework 2.2 In Class
3-12• p82-85
Homework 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
Symbolic Conditional Statements
To represent the hypothesis symbolically, we use the letter p. 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
If today is Tuesday, then tomorrow is Wednesday.
p = Today is Tuesday
q = Tomorrow is Wednesday
Symbolic form p q
• We read it to say “If p then q.”
Negation Recall that negation makes the
statement “negative.” 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 Converse
q p Inverse
~p ~q Contrapositive
~q ~p Biconditional
p q
Logical Argument 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 If pq is a true conditional statement and
p is true, then q is true. It should be stated to you that pq 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…
You will get credit for this class!
Law of Syllogism If pq and qr are true conditional
statements, then pr is true. 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.
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…
And this gets phrased using another conditional statement• If tomorrow is Wednesday, then there is a
quiz on Thursday.
Deductive v Inductive Reasoning Deductive reasoning
uses facts, definitions, and accepted properties in a logical order to write a proof.
This is often called a logical argument.
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.
Equivalent StatementsConditiona
lConverse 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.
Means “not”
If the conditional statement is true, then the contrapositive is also true. Therefore they are equivalent statements!
If the converse is true, then the inverse is also true. Therefore they are equivalent statements!
Homework In Class
1-5• p91-94
Homework 8-48 even
Due Tomorrow
Lesson 2.4
Reasoning withProperties ofAlgebra
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 Definition Identification Abbreviation
AdditionProperty
If a=b, then a+c = b+c.Something is added to both sides of the equation. APOE
SubtractionProperty
If a=b, then a-c = b-c.Something is subtracted from both sides of the equation.
SPOE
MultiplicationProperty
If a=b, then ac = bc.Something is multiplied to both sides of the equation. MPOE
DivisionProperty
If a=b and c≠0, thena/c = b/c.
Something is being divided into both sides. DPOE
SubstitutionProperty
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
DistributiveProperty
a(b+c) = ab + ac
A number outside of parentheses has been multiplied to all numbers inside.
DIST
Reflexive, Symmetric and Transitive Properties
Reflexive Symmetric Transitive
DefinitionFor any real number a,
a = a
If a=b, then b=a.
If a=b and b=c, then a=c.
Howto
Remember
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.
How to Use
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 This section is an introduction to proofs. To solve any algebra problem, you now
need to show ALL steps. 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 4Solve 9x+18=72
9x+18=72 GivenShort for “Information given to us.”
9x=54
x=6
-18 -18
SPOE
DPOE
9 9
Example 5: Using SegmentsIn the diagram, AB=CD. Show that AC=BD.
A B C D
AB=CD GivenThink about changing AB into AC? And the same with CD into BD?
AB+BC=BC+CD
AC=AB+BC
BD=BC+CD
AC=BD
APOE
Segment AdditionPostulate
Transitive POE
Segment AdditionPostulate
Example 6: Using Angles
HW Problem #24, p100 In the diagram, m RPQ=m RPS, verify to
show that m SPQ=2(m RPQ).
mRPQ=m RPS Given
m SPQ=m RPQ+m RPS
m SPQ=m RPQ+m RPQ
m SPQ=2(m RPQ)
Angle AdditionPostulate
SUB
DIST
S
R
Q
P
Example 7Fill in the two-column proof with the appropriate reasons for each step
APOE
MPOE
Symmetric POE
Homework 2.4 In Class
1,4-8• p99-101
Homework 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. 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
Prove the Symmetric Property of Segment Congruence. GIVEN: Segment PQ is congruent to Segment XY PROVE: Segment XY is congruent to Segment PQ
Hints for Making Proofs 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. 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 9Fill in the missing steps
Transitive POE
A C
Example 10Fill in the missing steps
1 and 2 are a linear pair
1 and 2 are supplementary
Definition of supplementary angles
m1 = 180o - m2
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
Theorem 2.1:Properties of Segment Congruence
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.
Theorem 2.2:Properties of Angle Congruence
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. 1 and 2 are right angles 1. Given
2. m1 = 90o, m2 = 90o 2. Definition of Right Angles
3. m1 = m2 3. Trans POE
4. 1 2 4. DEFCON
2
1
Theorem 2.4:Congruent Supplements Theorem
If two angles are supplementary to the same angle, or congruent angles, then they are congruent. If m1 + 2 = 180o and m2 + m3 = 180o,
then 1 3.
12
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 m4 + m5 = 90o and m5 + m6 =90o,
then 4 6.
54
6
Postulate 12:Linear Pair Postulate
The Linear Pair Postulate says if two angles form a linear pair, then they are supplementary.
1 2
1 + 2 = 180o
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.
12
34
1 3
2 4
Example 11Using the following figure, fill in the
missing steps to the proof.
Given
2
4
Definition of a linear pair
m1 + m2 = 180o
m3 + m4 = 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