Reasoning and Proof

1
Chapter 2

• Reasoning and Proof

2
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

3
Lesson 2.1

• Conditional Statements

4
Lesson 2.1 Objectives

• Analyze conditional statements
• Write postulates about points, lines, and planes
  using conditional statements

5
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.
6
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.
7
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.
8
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
9
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, cant, will not, nor,
  wouldnt, etc.

Today is Tuesday.
Today is not Tuesday.
10
Inverse

• 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.
11
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.
12
Point, Line, Plane PostulatesPostulate 5

• Through any two points there exists exactly one
  line.

13
Point, Line, Plane PostulatesPostulate 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.

14
Point, Line, Plane PostulatesPostulate 7

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

15
Point, Line, Plane PostulatesPostulate 8

• Through any three noncollinear points there
  exists exactly one plane.

16
Point, Line, Plane PostulatesPostulate 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.

17
Point, Line, Plane PostulatesPostulate 10

• If two points lie in a plane, then the line
  containing them lies in the same plane.

18
Point, Line, Plane PostulatesPostulate 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?

19
Homework 2.1

• In Class
• 1-8
• p75-78
• Homework
• 10-50 ev, 51, 55, 56
• Due Tomorrow

20
Lesson 2.2

• Definitions
• and
• Biconditional Statements

21
Lesson 2.2 Objectives

• Recognize a definition
• Recognize a biconditional statement
• Verify definitions using biconditional statements

22
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
23
Definition

• The previous slide was an example of a
  definition.
• It can be read forwards or backwards and maintain
  truth.

24
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.

25
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.

26
Example 1

• If ab 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.

27
Homework 2.2

• In Class
• 3-12
• p82-85
• Homework
• 14-42 even
• Due Tomorrow

28
Lesson 2.3

• Deductive Reasoning

29
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

30
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 ifthen, we use an arrow,
  ?.
• To represent the phrase if and only if, we use a
  two headed arrow, .

31
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.

32
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

33
Symbolic Variations

• Converse
• q ? p
• Inverse
• p ? q
• Contrapositive
• q ? p
• Biconditional
• p q

34
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
  ifthen 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.

35
Law of Detachment

• If p?q is a true conditional statement and p is
  true, then q is true.
• 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.

36
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!

37
Law of Syllogism

• If p?q and q?r are true conditional statements,
  then p?r 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.

38
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.

39
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.

40
Equivalent Statements
If the conditional statement is true, then the
contrapositive is also true. Therefore they are
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.
Means not
If the converse is true, then the inverse is also
true. Therefore they are equivalent statements!
41
Homework

• In Class
• 1-5
• p91-94
• Homework
• 8-48 even
• Due Tomorrow

42
Lesson 2.4

• Reasoning with
• Properties of
• Algebra

43
Lesson 2.4 Objectives

• Use properties from algebra to create a proof
• Utilize properties of length and measure to
  justify segment and angle relationships

44
Algebraic Properties of Equality
Property Definition Identification Abbreviation
Addition Property If ab, then ac bc. Something is added to both sides of the equation. APOE
Subtraction Property If ab, then a-c b-c. Something is subtracted from both sides of the equation. SPOE
Multiplication Property If ab, then ac bc. Something is multiplied to both sides of the equation. MPOE
Division Property If ab and c?0, then a/c b/c. Something is being divided into both sides. DPOE
Substitution Property If ab, then a can be substituted for b in any expression. One object is used in place of another without any calculations being done. SUB
Distributive Property a(bc) ab ac A number outside of parentheses has been multiplied to all numbers inside. DIST
45
Reflexive, Symmetric and Transitive Properties
Reflexive Symmetric Transitive
Definition For any real number a,a a If ab, then ba. If ab and bc, then ac.
How to 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.
46
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.

47
Example 4
Solve 9x1872
Short for Information given to us.
Given
9x1872
SPOE
9x54
x6
DPOE
48
Example 5 Using Segments
In the diagram, ABCD. Show that ACBD.
Think about changing AB into AC? And the same
with CD into BD?
ABCD
Given
ABBCBCCD
APOE
Segment Addition Postulate
ACABBC
Segment Addition Postulate
BDBCCD
Transitive POE
ACBD
49
Example 6 Using Angles

• HW Problem 24, p100
• In the diagram, m? RPQm? RPS, verify to show
  that m? SPQ2(m? RPQ).

m?RPQm? RPS
Given
Angle Addition Postulate
m? SPQm? RPQm? RPS
m? SPQm? RPQm? RPQ
SUB
DIST
m? SPQ2(m? RPQ)
50
Example 7

• Fill in the two-column proof with the appropriate
  reasons for each step

APOE
MPOE
Symmetric POE
51
Homework 2.4

• In Class
• 1,4-8
• p99-101
• Homework
• 10-32, 36-50 even
• Due Tomorrow

52
Lesson 2.5

• Proving Statements about Segments

53
Lesson 2.5 Objectives

• Write a two-column proof
• Justify statements about congruent segments

54
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.

55
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.

56
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

57
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!

58
Example 9

• Fill in the missing steps

Transitive POE
?A ? ?C
59
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
60
Homework 2.5

• In Class
• 1,3-5,7,9
• p105-107
• Homework
• 6-11,16,21,22
• Due Tomorrow

61
Lesson 2.6

• Proving Statements about Angles

62
Lesson 2.6 Objectives

• Utilize the angle and segment congruence
  properties
• Prove properties about special angle pairs

63
Theorem 2.1Properties 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.

64
Theorem 2.2Properties 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.

65
Theorem 2.3Right 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. Definition of Right Angles
2. m?1 90o, m?2 90o
3. m?1 m?2
3. Trans POE
4. ?1 ? ?2
4. DEFCON
66
Theorem 2.4Congruent 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.

2
1
3
67
Theorem 2.5Congruent 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.

5
4
6
68
Postulate 12Linear Pair Postulate

• The Linear Pair Postulate says if two angles form
  a linear pair, then they are supplementary.

?1 ?2 180o
1
2
69
Theorem 2.6Vertical Angles Theorem

• If two angles are vertical angles, then they are
  congruent.
• Vertical angles are angles formed by the
  intersection of two straight lines.

2
?1 ? ?3
1
3
4
?2 ? ?4
70
Example 11

• Using the following figure, fill in the missing
  steps to the proof.

Given
Definition of a linear pair
?2
?4
m?1 m?2 180o
m?3 m?4 180o
Congruent Supplements Theorem
71
Homework 2.6

• In Class
• 1,3-9,10,23
• p112-116
• Homework
• 10, 12-22, 27-28, 33-36
• Due Tomorrow