Chapter 2
Deductive Reasoning
• Learn deductive logic
• Do your first 2column proof
• New Theorems and
Postulates
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.
hypothesis
conclusion
If the sun is shining, then it is daytime.
Hidden If-Thens
A conditional may not contain either if or then!
All of my students love Geometry.
If you are my student, then you love Geometry.
Which is the hypothesis?
You are my student
Which is the conclusion?
you love Geometry
The Converse
A conditional with the hypothesis and
conclusion reversed.
Original: If the sun is shining, then it is daytime.
If q, then p.
hypothesis
conclusion
If it is daytime, then the sun is shining.
The Counterexample
If p, then q
FALSE
TRUE
The Counterexample
The only way a conditional can be false is if the
hypothesis is true and the conclusion is false. This
is called a counterexample.
The Counterexample
• This is HARD !
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
Group Practice
• Provide a counterexample to show that each
statement is false.
If you live in California, then you live in La
Crescenta.
Group Practice
• Provide a counterexample to show that each
statement is false.
If AB  BC, then B is the midpoint of AC.
Group Practice
• Provide a counterexample to show that each
statement is false.
If a line lies in a vertical plane, then the line is
vertical
Group Practice
• Provide a counterexample to show that each
statement is false.
If a number is divisible by 4, then it is
divisible by 6.
Group Practice
• Provide a counterexample to show that each
statement is false.
If x2 = 49, then x = 7.
Other Forms
•
•
•
•
If p, then q
p implies q
p only if q
q if p
What do you notice?
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
• 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
• 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
• Circle the hypothesis and underline the
conclusion
If a = b, then a + c = b + c
• Circle the hypothesis and underline the
conclusion
If a = b, then a + c = b + c
• Circle the hypothesis and underline the
conclusion
If a + c = b + c, then a = b
• Circle the hypothesis and underline the
conclusion
If a + c = b + c, then a = b
• 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
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.
If m1 = 90, then 1 is a right angle.
If 1 is a right angle, then m1 = 90.
m1 = 90 if and only if 1 is a right angle.
m1 = 90 iff 1 is a right angle.
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
Properties of Equality
Addition
Property
if x = y, then x + z = y + z.
Subtraction
Property
if x = y, then x – z = y – z.
Multiplication
Property
if x = y, then xz = yz.
Division
Property
if x = y, and z ≠ 0, then x/z = y/z.
Substitution
Property
Reflexive
Property
Symmetric
Property
Transitive
Property
if x = y, then either x or y may be
substituted for the other in any
equation.
x = x.
A number equals itself.
if x = y, then y = x.
Order of equality does not matter.
if x = y and y = z, then x = z.
Two numbers equal to the same
number are equal to each other.
Properties of Congruence
Reflexive
Property
Symmetric
Property
Transitive
Property
AB ≅ AB
A segment (or angle) is congruent to
itself
If AB ≅ CD, then CD ≅ AB
Order of equality does not matter.
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.
Your First Proof
Given: 3x + 7 - 8x = 22
Prove: x = - 3
1.
2.
3.
4.
STATEMENTS
3x + 7 - 8x = 22
-5x + 7 = 22
-5x = 15
x=-3
1.
2.
3.
4.
REASONS
Given
Substitution
Subtraction Prop. =
Division Prop. =
Your Second Proof
A
B
Given: AB = CD
Prove: AC = BD
1.
2.
3.
4.
C
D
STATEMENTS
REASONS
AB = CD
1. Given
AB + BC = BC + CD 2. Addition Prop. =
AB + BC = AC
3. Segment Addition Post.
BC + CD = BD
AC = BD
4. 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
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
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
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;
2. m  ABX = m  BXC
or
 ABX  m  BXC
3.m ABX + m BXC = m ABC
4.m ABX + m ABX = m ABC
or
2 m  ABX = m  ABC
5. m  ABX = ½ m  ABC
6. m  BXC = ½ m  ABC
C
1. Given
2. Definition of Angle Bisector
3. Angle Addition Postulate
4. Substitution
5. Division Property of Equality
6. Substitution Property
Reasons Used in Proofs
•
•
•
•
Given Information
Definitions
Postulates (including Algebra)
Theorems
How to write a proof
(The magical steps)
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,
remember my magical purple pen….use
yours. 
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
• 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
2.4 Special Pairs of Angles
Objectives
• Apply the definitions of complimentary and
supplementary angles
• State and apply the theorem about vertical
angles
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
B
C
X
T
See It!
Supplementary Angles
Any two angles whose measures sum to 180.
If mABC + m SXT = 180, then
 ABC and  SXT are supplementary. S
A
C
B
X
T
See It!
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
Theorem
Vertical angles are congruent
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 = 75
White Board Practice
• Find the measure of a complement and a
supplement of  T.
mT=a
White Board Practice
• Find the measure of a complement and a
supplement of  T.
m  T = 3x
White Board Practice
• Find the measure of a complement and a
supplement of  T.
m  T = 40
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
l
angles are right.
m
See It!
Theorem
If two lines are perpendicular, then they form
congruent, adjacent angles.
l
If l  m, then
1   2.
1
2
m
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
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
See It!
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
Theorem
If two angles are supplementary to congruent
angles (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
Practice
• Given: m  1 = m  4
Prove:  4 is supplementary to  2
1
2
3
4