Chapter 2
Deductive Reasoning
• Learn deductive logic
• Do your first 2column proof
• New Theorems and
Postulates
**PUT YOUR
LAWYER HAT ON!!
2.1 If – Then Statements
Objectives
• Recognize the hypothesis and conclusion of an ifthen statement
• State the converse of an if-then statement
• Use a counterexample
• Understand if and only if
The If-Then Statement
Conditional:is a two part statement with an actual or
implied if-then.
If p, then q.
p ---> q
hypothesis
conclusion
If I play football, then I am an athlete.
• Circle the hypothesis and underline the
conclusion
If a = b, then a + c = b + c
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?
two lines intersect
Which is the conclusion?
exactly one plane contains them
The whole thing:
If two lines intersect, then exactly one
plane contains them. (Theorem 1 – 3)
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.
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
If I am an athlete, then I play football.
**BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!
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…
The Counterexample
If p, then q
FALSE
TRUE
**You need only a single counterexample to prove a statement false.
The Counterexample
If x > 5, then x = 6.
x could be equal to 5.5 or 7 etc…
If x = 5, then 4x = 20
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.
White Board Practice
• Circle the hypothesis and underline the
conclusion
VW = XY implies VW  XY
• Circle the hypothesis and underline the
conclusion
VW = XY implies VW  XY
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
• Provide a counterexample to show that each
statement is false.
If a line lies in a vertical plane, then the line is
vertical
• Circle the hypothesis and underline the
conclusion
K is the midpoint of JL only if JK = KL
• Circle the hypothesis and underline the
conclusion
K is the midpoint of JL only if JK = KL
• Provide a counterexample to show that each
statement is false.
If a number is divisible by 4, then it is
divisible by 6.
• Provide a counterexample to show that each
statement is false.
If x2 = 49, then x = 7.
• Provide a counterexample to show that each
statement is false.
If AB  BC, then B is the midpoint of AC.
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
2.2 Properties from Algebra
Objectives
• Do your first proof
• Use the properties
of algebra and the
properties of
congruence in
proofs
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
Properties of Equality
Numbers, variables, lengths, and angle measures
WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO …
Addition
Property
Subtraction
Property
Add prop of
=
Subtr. Prop of
=
Multiplication
Property
Multp. Prop
of =
Division
Property
Div. Prop of
=
Substitution
Property
Substitution
Properties of Equality
Reflexive
Property
x = x.
A number equals itself.
Reflexive
Prop.
Transitive
Property
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
Transitive If AB ≅ CD and CD ≅ EF, then AB ≅ EF
Property
Two segments (or angles) congruent to
the same segment (or angle) are congruent to
each other.
Reflex.
Prop
Trans.
Prop
Rules of Thumb….
• Measurements are =
– (prop. of equality)
• Figures are 
– (Prop. of congruencey)
Whiteboards
• Page 40
– #’s 1 – 10
Your First Proof
Given: 3x + 7 - 8x = 22
Prove: x = - 3
(specifics)
(general rules)
STATEMENTS
1.
2.
3.
4.
3x + 7 - 8x = 22
-5x + 7 = 22
-5x = 15
x=-3
REASONS
1.
2.
3.
4.
Given
Substitution
Subtraction Prop. =
Division Prop. =
Day 2 - How to write a proof
• Walk-Thru of examples on page 38 and 39
Reasons Used in Proofs (pg. 45)
•
•
•
•
•
Given Information
Definitions (bi-conditional)
Postulates
Properties of equality and congruence
Theorems
Your Second Proof
Given : XZ = 20
YZ = 7
Prove: XY = 13
X
Y
Z
** Before we actually do this as a proof, lets make a verbal
argument about why this is true.**
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
Given : L2 = 50
Prove: L1 is congruent L3
3
1
2
Statements
Reasons
1.
1. given
2.
2.
3.
3.
4.
4.
Given : WX = YZ
Y is the midpoint of XZ
Prove: WX = XY
W
X
Y
** Before we actually do this as a proof, lets make a verbal
argument about why this is true.**
Z
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
Warm-up
• Page. 40 #12
• Discuss with class
2.3
Objectives
• Use the Midpoint
Theorem and the
Bisector Theorem
• Know the kinds of
reasons that can be
used in proofs
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!!
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?
The Angle Bisector Theorem
If BX is the bisector of ABC, then
m  ABX = ½ m  ABC
A
m  XBC = ½ m  ABC
X
B
C
Whiteboards
• Pg. 45 # 1-9
A
B
Given: AB = CD
Prove: AC = BD
STATEMENTS
1.
2.
3.
4.
AB = CD
BC = BC
AB + BC = BC + CD
AB + BC = AC
BC + CD = BD
5. AC = BD
C
D
REASONS
1. Given
2. Reflexive prop.
3. Addition Prop. =
4. Segment Addition Post.
5. Substitution
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)
WARM – UP
•
Answer true or false. If false, write a one
sentence explanation.
1. The converse of a true statement is sometimes
false.
2. Only one counterexample is needed to disprove
a statement.
3. Properties of equality cannot be used in
geometric proofs.
4. Postulates are deduced from theorems.
5. Every angle has only one bisector.
• Draw diagram on bottom of page 51 to
reference during lesson ( add a line to make
vertical angles)
2.4 Special Pairs of Angles
Objectives
• Apply the definitions of complimentary and
supplementary angles
• State and apply the theorem about vertical
angles
Complimentary & Supplementary
angles
• Rules that apply to either type..
1. We are always referring to a pair of
angles (2 angles) .. No more no less
2. Angles DO NOT have to be
adjacent
3. **Do not get confused with the
angle addition postulate
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
B
 SXT is the
complement of  ABC
C
X
T
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
 ABC is the
supplement of  SXT
 SXT is the
supplement of  ABC
C
B
X
T
Complimentary & Supplementary
angles
• Rules that apply to either type..
1. We are always referring to a pair of angles (2
angles) .. No more no less
2. Angles DO NOT have to be adjacent
3. **Do not get confused with the angle addition
postulate
4. 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!!
True or False
• m  A + m  B + m  C = 180, then
 A,  B, and  C are supplementary.
A- Sometimes
B – Always
C - Never
• Two right angles are ____________
complementary.
Vertical Angles
Two angles formed on the opposite sides of
the intersection of two lines.
1
4
2
3
Vertical Angles
Two angles formed on the opposite sides of
the intersection of two lines.
1
4
2
3
Vertical Angles
Two angles formed on the opposite sides of
the intersection of two lines.
1
4
2
3
The only thing the definition does is identify what vertical angles are…
NEVER USE THE DEFINITION IN A PROOF!!!
**THIS THEOREM
WILL BE USED IN
YOUR PROOFS OVER
AND OVER
Theorem
Vertical angles are congruent
(The definition of Vert. angles
does not tell us anything about congruency… this theorem proves that they are.)
1
4
2
3
White Board Practice
• Find the measure of a complement and a
supplement of  T.
m  T = 89
• If  1 and  2 are vertical angles,
m 1 = 2x+18 and m 2 = 3x+4, Find x.
•14
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
Whiteboard
Warm – Up
• Student will complete #33 from page 54 on
front board
2.5 Perpendicular Lines
Objectives
• Recognize
perpendicular lines
• Use the theorems
about perpendicular
lines
Perpendicular Lines ()
If two intersecting lines form right angles,
then they are perpendicular.
If l  m, then the
l
angles are right.
If the angles are
right, then l  m.
m
What can you conclude about the rest of the angles in
the diagram and why?
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 (biconditional)
– PG. 56
White Boards
• Page 57
– #1 , 4, 5
White Boards
Line AB  Line CD.
A
E
G
C
B
D
F
2.6 Planning a Proof
Objectives
• Discover the steps used to plan a proof
Practice
• Given: m  1 = m  4
Prove: m 4 + m 2 = 180
1
2
3
4
`
Statements
Reasons
1. mL1 = m L4
1. Given
2.
2.
3.
3.
4.
4.
Practice
• Given: m 2 + m 3 = 180
Prove: m  1 = m  3
1
2
3
4
• Given: x  m
• Prove: mL1 + mL2 = 90
x
C
1
B
2
m
A