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Chapter 2Deductive Reasoning
• Learn deductive logic• Do your first 2-
column proof• New Theorems and
Postulates
**PUT YOUR LAWYER HAT ON!!
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
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.
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
exactly one plane contains them
The whole thing:If two lines intersect, then exactly one
plane contains them. (Theorem 1 – 3)
All theorems, postulates, and definitions are conditional statements!!
The Converse
A conditional with the hypothesis and conclusion
reversed.
If q, then p. q ---> p
hypothesis conclusion
If it is daytime, then the sun is shining.
Original: If the sun is shining, then it is daytime.
**BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!
• 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
The Counterexample
If p, then q
TRUEFALSE
**You need only a single counterexample to prove a statement false.
The Counterexample If x > 5, then x = 6.
If x = 5, then 4x = 20
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.
Other Forms
• If p, then q
• p implies q
• p only if q
• q if p
What do you notice?
Conditional statements are not always written with the “if” clause first.
All of these conditionals mean the same thing.
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.
The Biconditional
Statement: If m1 = 90, then 1 is a right angle.
Converse: If 1 is a right angle, then m1 = 90.
m1 = 90 if and only if 1 is a right angle.
1 is a right angle if and only if m1 = 90 .
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
• 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 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
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
=
Numbers, variables, lengths, and angle measures
WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO …
Substitution
Property
if x = y, then either x or y may be substituted for the other in any equation.
Substitution
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 = 22Prove: x = - 3
1. 3x + 7 - 8x = 22 1. Given
2. -5x + 7 = 22 2. Substitution
3. -5x = 15 3. Subtraction Prop. =
4. x = - 3 4. Division Prop. =
(specifics) (general rules)
STATEMENTS REASONS
Properties of Congruence
Reflexive
Property
AB AB≅
A segment (or angle) is congruent to itself
Reflex.
Prop
Symmetric Property
If AB CD, then CD AB≅ ≅
Order of equality does not matter.
Symm.
Prop
Transitive
Property
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.
Prop
Segments, angles and polygons
Your Second Proof
Given: AB = CDProve: AC = BD
1. AB = CD 1. Given
2. BC = BC 2. Reflexive prop.
3. AB + BC = BC + CD 3. Addition Prop. =
4. AB + BC = AC 4. Segment Addition Post.
BC + CD = BD
5. AC = BD 5. Substitution
STATEMENTS REASONS
A
BC
D
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
• 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?
Statements (specifics) Reasons (general rules)
1. M is the midpoint of AB 1. Given2. AM MB or AM = MB 2. Definition of a midpoint3. AM + MB = AB 3. Segment Addition Postulate4. AM + AM = AB 4. Substitution Property Or 2 AM = AB5. AM = ½ AB 5. Division Property of Equality6. MB = ½ AB 6. Substitution
Given: M is the midpoint of ABProve: AM = ½ AB and MB = ½ AB
A M B
The Angle Bisector Theorem
If BX is the bisector of ABC, then
m ABX = ½ m ABC
m XBC = ½ m ABC A
B
X
C
1. BX is the bisector of ABC; 1. Given2. m ABX = m XBC 2. Definition of Angle Bisector or ABX m XBC3.m ABX + m XBC = m ABC 3. Angle Addition Postulate4.m ABX + m ABX = m ABC 4. Substitution
or 2 m ABX = m ABC5. m ABX = ½ m ABC 5. Division Property of =6. m XBC = ½ m ABC 6. Substitution Property
Given: BX is the bisector of ABCProve: m ABX = ½ m ABC m XBC = ½ m ABC
A
B
X
C
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.
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
• 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 Reasons1. 1.2. 2.3. 3.4. 4.Etc… Etc…
Statements Reasons
1. m 1 = m 2;
AD bisects CAB;
BD bisects CBA
1. Given
2. m 1 = m 3;
m 2 = m 4
2. Def of bisector
3. m 3 = m 4 3. Substitution
Statements Reasons
1. WX = YZ
Y is the midpoint of XZ
1. Given
2. XY = YZ 2. Def of midpoint
3. WX = XY 3. 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)
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 AnglesAny two angles whose measures add up to 90.
If mABC + m SXT = 90, then
ABC and SXT are complimentary.
S
X
T
A
B C
ABC is the complement of SXT SXT is the complement
of ABC
Supplementary AnglesAny two angles whose measures sum to 180.
If mABC + m SXT = 180, then
ABC and SXT are supplementary.
S
X
T
A
BC
ABC is the supplement of SXT SXT is the
supplement of ABC
Vertical AnglesTwo angles formed on the opposite sides of
the intersection of two lines.
1
2
3
4
The only thing the definition does is identify what vertical angles are…NEVER USE THE DEFINITION IN A PROOF!!!
Theorem
Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency… this theorem proves that they are.)
1
2
3
4
**THIS THEOREM WILL BE USED IN
YOUR PROOFS OVER AND OVER
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
angles are right.l
m
If the angles are right, then l m.
• 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 (bi-conditional)– PG. 56
Perpendicular Lines ()
TheoremIf two lines are perpendicular, then they form
congruent, adjacent angles.
l
m1 2
If l m, then
1 2.
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
m1 2
If 1 2, then
l 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?
TheoremIf the exterior sides of two adjacent angles lie on
perpendicular lines, then the angles are complimentary.
m
l
12
If l m, then
1 and 2 are compl.
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:
Construct:
Steps:
ABbisector of AB
Construction 5
Given a point on a line, construct the perpendicular to the line through the point.
Given:
Construct:
Steps:
line l with point A to l through A
Construction 6
Given a point outside a line, construct the perpendicular to the line through the point.
Given:
Construct:
Steps:
line l with point A to l through A
TheoremIf 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
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
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
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