1. Chapter 2 Deductive Reasoning
Learn deductive logic
Do your first 2-column proof
New Theorems and Postulates
**PUT YOUR LAWYER HAT ON!!
In algebra, the emphasis is on solving problems and getting right answers. In Geometry, the emphasis is on how to solve problems, but not the answer. In a proof, we already know the answer. The purpose of a proof is to show why the answer is true.

2. 2.1 If – Then Statements Objectives
Recognize the hypothesis and conclusion of an if-then statement
State the converse of an if-then statement
Use a counterexample
Understand if and only if
The if-then statement forms the basis for a syllogistic argument, which is the logical form used in a proof.

3. The If-Then Statement Conditional:
is a two part statement with an actual or implied if-then.
If p, then q.
p ---> q
Logicians replace the phrases in an argument with letters to emphasize the form of the argument, rather than its meaning. Whether an argument is valid or invalid has nothing to do with the meaning, and with the words stripped away, this is easier to see. Once words are inserted into a conditional, we start to attach meaning to them, and try to evaluate the validity of the argument based on the meaning of the words: is it true or false? Truth or falsity have nothing to do with validity, so arguments are better left in symbols when validity is being examined.
You use if-then statements to form a chain of logical deductions
hypothesis
conclusion
If I play football, then I am an athlete.

4. Circle the hypothesis and underline the conclusion
If a = b, then a + c = b + c 4

5. Hidden If-Thens A conditional may not contain either if or then!
All theorems, postulates, and definitions are conditional statements!!
Hidden If-Thens
A conditional may not contain either if or then!
Two intersecting lines are contained in exactly one plane.
Which is the hypothesis?
Which is the conclusion?
two lines intersect
All theorems, postulates, properties and definitions are conditionals. So is the previous sentence.
exactly one plane contains them
The whole thing:
If two lines intersect, then exactly one plane contains them. (Theorem 1 – 3)

6. Other Forms If p, then q p implies q p only if q
Conditional statements are not always written with the “if” clause first.
All of these conditionals mean the same thing. 6

7. Definition of Converse
A conditional with the hypothesis and conclusion
reversed.
Original: If the sun is shining, then it is daytime.
If q, then p.
q ---> p
hypothesis
conclusion
Many students confuse the converse with the conditional, and assume that the logic runs both ways. A careful analysis will keep them from making that mistake.
A statement and its converse can say different things. Some true statements have false converses.
If I am an athlete, then I play football.
**BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!

8. Definition of Counterexample
Using the same hypothesis as the statement, but coming to a different conclusion.
*Like a lawyer providing an alibi for his client…
Definitions, Theorems and postulates have no counterexample. Otherwise they would not be true. A conditional with only a single counterexample is false. To be true, it must always be true, with no exceptions.

9. The Counterexample If p, then q FALSE TRUE
Definitions, Theorems, and postulates have no counterexample. Otherwise they would not be true. A conditional with only a single counterexample is false. To be true, it must always be true, with no exceptions.
**You need only a single counterexample to prove a statement false.

10. The Counterexample If x > 5, then x = 6.
x could be equal to 5.5 or 7 etc…
always true, no counterexample
**Definitions, Theorems and postulates have no counterexample. Otherwise they would not be true. To be true, it must always be true, with no exceptions.
Theorems and postulates have no counterexample. Otherwise they would not be true. A conditional with only a single counterexample is false. To be true, it must always be true, with no exceptions.

11. White Board Practice
Circle the hypothesis and underline the conclusion
VW = XY implies VW  XY

12. Circle the hypothesis and underline the conclusion
VW = XY implies VW  XY

13. Write the converse of each statement
If I play the tuba, then I am in the band.
If I am in the band, then I play tuba.
If 2x = 4, then x = 2
If x = 2, then 2x = 4

14. Provide a counterexample to show that each statement is false.
If a line lies in a vertical plane, then the line is vertical

15. Circle the hypothesis and underline the conclusion
K is the midpoint of JL only if JK = KL

16. Circle the hypothesis and underline the conclusion
K is the midpoint of JL only if JK = KL

17. Provide a counterexample to show that each statement is false.
If a number is divisible by 4, then it is divisible by 6. 17

18. Provide a counterexample to show that each statement is false.
If x2 = 49, then x = 7. 18

19. Provide a counterexample to show that each statement is false.
If AB  BC, then B is the midpoint of AC. 19

20. If 3 points are in line, then they are colinnear.
WARM UP Is the original statement T or F? Then write the converse… if false, provide a counter example.
If 3 points are in line, then they are colinnear.
If 3 points are colinnear, then they are in line.
If I live in Los Angeles, then I live in CA.
If I live in CA, then I live in Los Angleles.
False, you could live in San Diego

21. 2.2 Properties from Algebra
Objectives
Do your first proof
Use the properties of algebra and the properties of congruence in proofs
Proofs baffle most geometry students at the start. One of the best ways to ensure that they make the connection between the given and the prove is to link it to something that they know: algebra.

22. Properties from Algebra
see properties on page 37
Read the first paragraph
This lesson reviews the algebraic properties of equality that will be used to write proofs and solve problems.
We treat the properties of Algebra like postulates
Meaning we assume them to be true
Go through the properties of equality and congruency and talk about what each one means. Indicate that they already know how to solve problems with these properties: they do so every time they solve an algebraic equation. Now they are going to use these properties like postulates and use them to prove why an equation has the solution that it does.

23. Properties of Equality
Numbers, variables, lengths, and angle measures
WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO …
Addition
Property
Add prop of =
Subtraction Property
Subtr. Prop of =
Multiplication Property
Multp. Prop of =
Division
Div. Prop of
Substitution
Come up with examples based on algebraic problems that you have done in the past
When you write these in your proofs, you will write “________ property of =“ (which explains that it is a property of equality)

24. Properties of Equality
Reflexive
Property
x = x.
A number equals itself.
Reflexive Prop.
Transitive
if x = y and y = z, then x = z.
Two numbers equal to the same number are equal to each other.
Transitive Pop.
Properties of Congruence
Reflexive
Property
AB ≅ AB
A segment (or angle) is congruent to itself
Reflex.
Prop
Transitive
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
Two segments (or angles) congruent to
the same segment (or angle) are congruent to each other.
Trans.
When talking about substitution property explain how (5x – 2x) is the same as 3x
Just write… “substitution”
The last three properties highlighted in yellow lead to the three properties of congruence.

25. Rules of Thumb…. Measurements are = Figures are  (prop. of equality)
(Prop. of congruencey)

26. Whiteboards
Page 40
#’s 1 – 10

27. Your First Proof Given: 3x + 7 - 8x = 22 Prove: x = - 3
(specifics) (general rules)
STATEMENTS REASONS
1. 3x x = Given
x + 7 = Substitution
x = Subtraction Prop. =
4. x = Division Prop. =
Take this very slowly. Have them look at the structure and format of the proof using this example. Note that the answer is already provided: doing a proof has nothing to do with finding the answer, it is all about justifying the answer logically.

28. Day 2 - How to write a proof
Walk-Thru of examples on page 38 and 39

29. Reasons Used in Proofs (pg. 45)
Given Information
Definitions (bi-conditional)
Postulates
Properties of equality and congruence
Theorems
They should begin organizing the statements they learn in class every day by making 3x5 note cards with the significant theorems, postulates, properties and definitions on them.

30. Your Second Proof Given : XZ = 20 YZ = 7 Prove: XY = 13 Y Z X
** Before we actually do this as a proof, lets make a verbal argument about why this is true.**

31. Statements
Reasons 1. 2. 3. 4.

32. Given : L2 = 50
Prove: L1 is congruent L3 3 2 1

33. Statements
Reasons 1.
1. given 2. 3. 4.

34. Given : WX = YZ Y is the midpoint of XZ Prove: WX = XY Y Z W X
** Before we actually do this as a proof, lets make a verbal argument about why this is true.**

35. Statements
Reasons
1. Y is the midpoint of XZ
1. Given
2. XY = YZ
2. Def of midpoint
3. WX = YZ
3. Given
4. WX = XY
4. Substitution

36. Warm-up
Page. 40 #12
Discuss with class

37. 2.3 Objectives Use the Midpoint Theorem and the Bisector Theorem
Know the kinds of reasons that can be used in proofs

38. Being a lawyer…
When making your case, you might reference laws, statutes, and/or previous cases in order to make your argument…
YOU BETTER MAKE SURE YOU ARE REFERECING THE CORRECT ONES OR THE JUDGE WILL KICK YOU OUT OF THE COURTROOM!!

39. The Midpoint Theorem If M is the midpoint of AB, then
AM = ½ AB and MB = ½ AB
How is the definition of a midpoint different from this theorem?
One talks about congruent segments
One talks about something being half of something else
How do you know which one to use in a proof?
This is the first theorem that can be proven with a 2-column proof. Show them how to extract the given and prove from the conditional statement that is the theorem. Mark the figure, take an inventory and do the proof.
Distinguish between the Definition of a Midpoint and this theorem. They both talk about a midpoint, and are often confused. What is their difference? (one talks about congruent segments, while the other talks about something being half of something else) How would you know which one to use in a proof? (Does the proof feature fractions of one-half?)

40. The Angle Bisector Theorem
If BX is the bisector of ABC, then
m  ABX = ½ m  ABC
m  XBC = ½ m  ABC A
Ditto. Does this theorem sound familiar? Leave the other proof on the board and have them prove this one on their own. X B C

41. Whiteboards
Pg. 45 # 1-9

42. 3. AB + BC = BC + CD 3. Addition Prop. =
Given: AB = CD C
Prove: AC = BD D
STATEMENTS REASONS
1. AB = CD Given
2. BC = BC Reflexive prop.
3. AB + BC = BC + CD Addition Prop. =
4. AB + BC = AC Segment Addition Post.
BC + CD = BD
5. AC = BD Substitution
Also take your time with this one. Show them to mark the given on the figure. Have them look at the figure and the given and put into words why this conclusion is true. If they can see it as true, they have already done the proof in their heads. To write the proof, they have to slow down their thought process and discover what the connections are between the given and the prove line. Have them write down all the postulates and properties that they have learned that might help with this proof (SAP, Addition Property of Equality, Substitution) and then help them to see how to place the statements and reasons so the whole is linked together from start (given) to finish (prove).

43. QUIZ REVIEW Underline the hypothesis and conclusion in each statement
Write a converse of each statement and tell whether it is true or false
Provide a counter example to show that the statement is false
Be able to complete a proof
Name the reasons used in a proof (there are 5)

44. WARM – UP
Answer true or false. If false, write a one sentence explanation.
The converse of a true statement is sometimes false.
Only one counterexample is needed to disprove a statement.
Properties of equality cannot be used in geometric proofs.
Postulates are deduced from theorems.
Every angle has only one bisector.

45. Draw diagram on bottom of page 51 to reference during lesson ( add a line to make vertical angles)

46. 2.4 Special Pairs of Angles
Objectives
Apply the definitions of complimentary and supplementary angles
State and apply the theorem about vertical angles

47. Complimentary & Supplementary angles
Rules that apply to either type..
We are always referring to a pair of angles (2 angles) .. No more no less
Angles DO NOT have to be adjacent
**Do not get confused with the angle addition postulate

48. Definition :Complimentary Angles
If two angles add up to 90, then they are complimentary.
If mABC + m SXT = 90, then
 ABC and  SXT are complimentary. S A
 ABC is the complement of  SXT
 SXT is the complement of  ABC
Complimentary angles do not have to be adjacent. This is the major difference between this definition and the AAP. To use the AAP, the angles have to be adjacent. X C B T

49. Definition: Supplementary Angles
If two angles add up to 180, then the angles are supplementary.
If mABC + m SXT = 180, then
 ABC and  SXT are supplementary. S A
Ditto.
 ABC is the supplement of  SXT
 SXT is the supplement of  ABC X C B T

50. Complimentary & Supplementary angles
Rules that apply to either type..
We are always referring to a pair of angles (2 angles) .. No more no less
Angles DO NOT have to be adjacent
**Do not get confused with the angle addition postulate
In proofs, you must first prove two L’s add up to 90 or 180 before saying they are comp or suppl. NEED TO BE EXPLICT!!

51. True or False
m  A + m  B + m  C = 180, then  A,  B, and  C are supplementary.

52. A- Sometimes B – Always C - Never
Two right angles are ____________ complementary.

53. Vertical Angles
Two angles formed on the opposite sides of the intersection of two lines. 1
This is just the definition. It does not say anything useful, other than identify what vertical angles are. Are vertical angles congruent? Why? While the definition does not say they are congruent, there is a theorem that does… 4 2 3

54. Vertical Angles
Two angles formed on the opposite sides of the intersection of two lines. 1
This is just the definition. It does not say anything useful, other than identify what vertical angles are. Are vertical angles congruent? Why? While the definition does not say they are congruent, there is a theorem that does… 4 2 3

55. Vertical Angles
Two angles formed on the opposite sides of the intersection of two lines. 1 4 2 3
This is just the definition. It does not say anything useful, other than identify what vertical angles are. Are vertical angles congruent? Why? While the definition does not say they are congruent, there is a theorem that does…
The only thing the definition does is identify what vertical angles are…
NEVER USE THE DEFINITION IN A PROOF!!!

56. **THIS THEOREM WILL BE USED IN YOUR PROOFS OVER AND OVER
Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency… this theorem proves that they are.) 1
Do a proof of this theorem, if there is time. 4 2 3

57. White Board Practice
Find the measure of a complement and a supplement of  T.
m  T = 89

58. If  1 and  2 are vertical angles, m 1 = 2x+18 and m 2 = 3x+4, Find x.14

59. White Board Practice
A supplement of an angle is three times as large as a complement of the angle. Find the measure of the angle.
Let x = the measure of the angle.
180 – x : This is the supplement
90 – x : This is the complement
180 – x = 3 (90 – x)
180 – x = 270 – 3x
2x = 90
x = 45

60. Whiteboard

61. Warm – Up
Student will complete #33 from page 54 on front board

62. 2.5 Perpendicular Lines Objectives Recognize perpendicular lines
                                   
Objectives
Recognize perpendicular lines
Use the theorems about perpendicular lines

63. Perpendicular Lines ()
If two intersecting lines form right angles, then they are perpendicular.
If l  m, then the
angles are right. l
Notice that the definition does not say anything about 90. It takes the definition of a right angle to do that.
If the angles are right, then l  m. m
What can you conclude about the rest of the angles in the diagram and why?

64. Perpendicular Lines ()
Two lines that form one right angle form four right angles
The definition applies to intersecting rays and segments
The definition can be used in two ways (bi-conditional)
PG. 56

65. White Boards
Page 57
#1 , 4, 5

66. White Boards
Line AB  Line CD. A E G C D B F

67. 2.6 Planning a Proof Objectives
Discover the steps used to plan a proof

68. Practice
Given: m  1 = m  4
Prove: m 4 + m 2 = 180 3 1 2 4

69. `
Statements
Reasons
1. mL1 = m L4
1. Given 2. 3. 4.

70. Practice
Given: m 2 + m 3 = 180
Prove: m  1 = m  3 3 1 2 4

71. Given: x  m
Prove: mL1 + mL2 = 90 x C 1 2 B m A