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 the sun is shining, then it is daytime.
• 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)
The 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 it is daytime, then the sun is shining.
**BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!
The Counterexample
• An if –then statement is false if an
example can be found for which the
hypothesis is true and the conclusion is
false.
• The example is called the
Counterexample.
• *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.
Other Forms
•
•
•
•
If p, then q
p implies q
p only if q
q if p
Conditional statements are not
always written with the “if”
clause first.
All of these conditionals mean
the same thing.
What do you notice?
The Biconditional
If a conditional and its converse are the same
(both true) then it is a biconditional and can
use the “if and only if” language.
Statement: If m1 = 90, then 1 is a right angle.
Converse: If 1 is a right angle, then m1 = 90.
m1 = 90 if and only if 1 is a right angle.
1 is a right angle if and only if m1 = 90 .
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 football, then I am an athlete
• If I am an athlete, then I play football
• 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.
• Circle the hypothesis and underline the
conclusion
n > 8 only if n is greater than 7
• Circle the hypothesis and underline the
conclusion
n > 8 only if n is greater than 7
• Circle the hypothesis and underline the
conclusion
I’ll dive if you dive
• Circle the hypothesis and underline the
conclusion
I’ll dive if you dive
• Provide a counterexample to show that each
statement is false.
If x2 = 49, then x = 7.
• Circle the hypothesis and underline the
conclusion
r + n = s + n if r = s
• Circle the hypothesis and underline the
conclusion
r + n = s + n if r = s
• Provide a counterexample to show that each
statement is false.
If AB  BC, then B is the midpoint of AC.
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
if x = y, then x + z = y + z.
Add prop of
=
Subtraction
Property
if x = y, then x – z = y – z.
Subtr. Prop of
=
Multiplication
Property
if x = y, then xz = yz.
Multp. Prop
of =
Division
Property
if x = y, and z ≠ 0, then x/z = y/z.
Div. Prop of
=
Substitution if x = y, then either x or y may be
Substitution
substituted for the other in any equation.
Property
Reflexive
Property
x = x.
A number equals itself.
Reflexive
Prop.
Symmetric
Property
if x = y, then y = x.
Order of equality does not matter.
Symmetric
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.
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. =
Properties of Congruence
Segments, angles and polygons
Reflexive
Property
AB ≅ AB
A segment (or angle) is congruent to itself
Reflex.
Prop
Symmetric If AB ≅ CD, then CD ≅ AB
Property
Order of equality does not matter.
Symm.
Prop
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.
Trans.
Prop
Your Second Proof
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
2.3 Proving Theorems
Objectives
• Use the Midpoint
Theorem and the
Bisector Theorem
• Know the kinds of
reasons that can be
used in proofs
PB & J Sandwich
• How do I make one?
– Pretend as if I have never made a PB & J
sandwich. Not only have I never made one, I
have never seen one or heard about a sandwich
for that matter.
– Write out detailed instructions in full sentences
– I will collect this
First, open the bread package by
untwisting the twist tie. Take out
two slices of bread set one of these
pieces aside. Set the other in front of
you on a plate and remove the lid
from the container with the peanut
butter in it.
Take the knife, place it in the
container of peanut butter, and with
the knife, remove approximately a
tablespoon of peanut butter. The
amount is not terribly relevant, as
long as it does not fall off the knife.
Take the knife with the peanut butter
on it and spread it on the slice of
bread you have in front of you.
Repeat until the bread is reasonably
covered on one side with peanut
butter. At this point, you should wipe
excess peanut butter on the inside
rim of the peanut butter jar and set
the knife on the counter.
Replace the lid on the peanut butter
jar and set it aside. Take the jar of
jelly and repeat the process for
peanut butter. As soon as you have
finished this, take the slice of bread
that you set aside earlier and place it
on the slice with the peanut butter
and jelly on it, so that the peanut
butter and jelly is reasonably well
contained within.
The Midpoint Theorem
If M is the midpoint of AB, then
AM = ½ AB and MB = ½ AB
A
M
B
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?
Important Notes
• Does the order matter?
• Don’t leave out steps  Don’t Assume
Given: M is the midpoint of AB
Prove: AM = ½ AB and MB = ½ AB
A
Statements (specifics)
1.
2.
3.
4.
M is the midpoint of AB
AM  MB or AM = MB
AM + MB = AB
AM + AM = AB
Or
2 AM = AB
5. AM = ½ AB
6. MB = ½ AB
M
B
Reasons (general rules)
1.
2.
3.
4.
Given
Definition of a midpoint
Segment Addition Postulate
Substitution Property
5. Division Property of Equality
6. Substitution
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
A
Given: BX is the bisector of ABC
Prove: m  ABX = ½ m  ABC
m  XBC = ½ m  ABC
X
B
1. BX is the bisector of  ABC;
1.
2. m  ABX = m  XBC
2.
or
 ABX  m  XBC
3.m ABX + m  XBC = m ABC 3.
4.m ABX + m ABX = m ABC 4.
or
2 m  ABX = m  ABC
5. m  ABX = ½ m  ABC
5.
6. m  XBC = ½ m  ABC
6.
C
Given
Definition of Angle Bisector
Angle Addition Postulate
Substitution
Division Property of =
Substitution Property
Reasons Used in Proofs (pg. 45)
•
•
•
•
•
Given Information
Definitions (bi-conditional)
Postulates
Properties of equality and congruence
Theorems
How to write a proof
(The magical steps)
• Use these steps every time you have to do a
proof in class, for homework, on a test, etc.
Example 1
Given : m  1 = m  2;
AD bisects  CAB;
BD bisects  CBA
Prove: m  3 = m  4
D
1
A
C
2
3
4
B
1. Copy down the problem.
• Write down the given and prove statements
and draw the picture. Do this every single
time, I don’t care that it is the same picture,
or that the picture is in the book.
– Draw big pictures
– Use straight lines
2. Mark on the picture
• Read the given information and, if possible,
make some kind of marking on the picture.
• Remember if the given information doesn’t
exactly say something, then you must think
of a valid reason why you can make the
mark on the picture.
• Use different colors when you are marking
on the picture.
3. Look at the picture
• This is where it is really important to know
your postulates and theorems. Look for
information that is FREE, but be careful
not to Assume anything.
–
–
–
–
Angle or Segment Addition Postulate
Vertical angles
Shared sides or angles
Parallel line theorems
4. Brain.
• Do you have one?
• I mean have you drawn a brain and are you
writing down your thought process? Every
single time you make any mark on the
picture, you should have a specific reason
why you can make this mark. If you can do
this, then when you fill the brain the proof
is practically done.
5. Finally look at what you are
trying to prove
• Ask yourself: “Does it make sense?”
“why?”
• Write out a plan to help organize your
thoughts
• Then try to work backwards and fill in any
missing links in your brain. Think about
how you can get that final statement.
6. Write the proof.
• (This should be the easy part)
Statements
1.
2.
3.
4.
Etc…
Reasons
1.
2.
3.
4.
Etc…
Example 1
Given : m  1 = m  2;
AD bisects  CAB;
BD bisects  CBA
Prove: m  3 = m  4
D
1
A
C
2
3
4
B
Statements
Reasons
1. m 1 = m  2;
AD bisects  CAB;
BD bisects  CBA
1. Given
2. m 1 = m  3;
m 2 = m  4
3. m 3 = m  4
2. Def of  bisector
3. Substitution
Try it
Given : WX = YZ
Y is the midpoint of XZ
Prove: WX = XY
W
X
Y
Z
Statements
Reasons
1. WX = YZ
Y is the midpoint of
XZ
2. XY = YZ
1. Given
3. WX = XY
3. Substitution
2. Def of midpoint
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)
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
Complimentary Angles
Any two angles whose measures add up to 90.
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
Supplementary Angles
Any two angles whose measures sum to 180.
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
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
Remote Time
True or False
• m  A + m  B + m  C = 180, then ,
 B, and  C are supplementary.
True or False
• Vertical angles have the same measure
True or False
• If  1 and  2 are vertical angles, m  1 =
2x+18, and m  2 = 3x+4, then x = 14.
A- Sometimes
B – Always
C - Never
• Vertical angles ____________ have a
common vertex.
A- Sometimes
B – Always
C - Never
• Two right angles are ____________
complementary.
A- Sometimes
B – Always
C - Never
• Right angles are ___________ vertical
angles.
A- Sometimes
B – Always
C - Never
• Angles A, B, and C are __________
complementary.
A- Sometimes
B – Always
C - Never
• Vertical angles ___________ have a
common supplement.
White Board Practice
• Find the measure of a complement and a
supplement of  T.
m  T = 40
White Board Practice
• Find the measure of a complement and a
supplement of  T.
m  T = 89
White Board Practice
• Find the measure of a complement and a
supplement of  T.
mT=a
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
2.5 Perpendicular Lines
Objectives
• Recognize
perpendicular lines
• Use the theorems
about perpendicular
lines
Perpendicular Lines ()
Two lines that intersect to form right angles.
If l  m, then the
l
angles are right.
If the angles are
right, then l  m.
m
Perpendicular Lines ()
• Two lines that form one right angle form four
right angles
• You can conclude that two lines are perpendicular
by definition, once you know that any of the
angles they form is a right angle
• The definition applies to intersecting rays and
segments
• The definition can be used in two ways (biconditional)
– PG. 56
Theorem
If two lines are perpendicular, then they form
congruent, adjacent angles.
l
If l  m, then
1   2.
1
2
m
PARTNERS: Complete
the proof on page 57
problem #1
Theorem
If two lines intersect to form congruent, adjacent
angles, then the lines are perpendicular.
l
If 1   2, then
l  m.
1
2
m
Partners
THINK – PAIR – SHARE
• Discuss the wording of Theorems 2 – 4 and
2 – 5.
• Look at the hypothesis and conclusion of
each
• When would you use each in a proof?
Theorem
If the exterior sides of two adjacent angles lie
on perpendicular lines, then the angles are
complimentary.
l
If l  m, then
1 and  2 are compl.
1
2
m
CAN ANYONE EXPLAIN?
PARTNERS
• Answer questions 6-10 on page 57
• #6 – Def. of perp. lines
• #7 – Def. of perp. Lines
• #8 – If 2 lines are perp., then they form cong. Adj.
angles
• #9 – Def. of perp. Lines
• #10 – IF 2 lines form cong. Adj. angles, then the
lines are perp.
Construction 4
Given a segment, construct the perpendicular bisector
of the segment.
Given: AB
Construct:
 bisector of
Steps:
AB
Construction 5
Given a point on a line, construct the perpendicular to
the line through the point.
Given: line l with point A
Construct:  to l through A
Steps:
Construction 6
Given a point outside a line, construct the
perpendicular to the line through the point.
Given: line l with point A
Construct:  to l through A
Steps:
2.6 Planning a Proof
Objectives
• Discover the steps used to plan a proof
Remember Magical Proof Steps
Demo
• Complimentary and supplementary
Theorems
• I need 4 Volunteers
Theorem
If two angles are supplementary to congruent
angles (or the same angle) then they are
congruent.
If 1 suppl  2 and  2 suppl  3, then
 1   3.
1
2
3
Theorem
If two angles are complimentary to congruent
angles (or to the same angle) then they are
congruent.
If 1 compl  2 and  2 compl  3, then
 1   3.
1
2
3
Practice
• Given:  2 and  3 are supplementary
Prove: m  1 = m  3
1
2
3
4
Statements
Reasons
1. L2 and L3 are supp.
1. Given
2. mL2 +m L1 = 180
2. angle addition
postulate
3. L2 is supp. to L1
3. Def of supp. angles
4. mL1 = mL3
4.If two angles are supp. to the same
angle, then the two angles are congruent
Practice
• Given: m  1 = m  4
Prove:  4 is supplementary to  2
1
2
3
4
Statements
Reasons
1. mL1 = m L4
1. Given
2. mL2 +m L1 = 180
2. angle addition
postulate
3. L2 is supp. to L1
3. Def of supp. angles
4. L4 is supp. to L2
4.Substituion
Test Review
• Underline the hypothesis and conclusion in each statement
• Write a converse of each statement and tell whether it is
true or false
• Fill in the blanks of an algebraic proof and a geometric
proof
• Name the following
– Complementary / supplementary angles
– Perpendicular lines or rays
– Vertical angles
• Understand what you can deduce from a
diagram that is marked
• Right angles = 90 / Straight angles that = 180
• Using vertical angles to find measures
• Setting up an algebraic problem of = angles in order
to solve for a variable – then using the variable to
solve the measure of other angles
• **SHOWING YOUR WORK IN THE ANSWER
DOCUMENT WHEN SOLVING FOR A
VARIABLE OR MEASUREMENT**
• Setting up and solving an equation
involving a supplement and complement of
an angle
• Complete an entire geometric proof