Chapter 2 Deductive Reasoning Learn deductive logic Do your first 2- column proof New Theorems and...

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.

• Circle the hypothesis and underline the conclusion

If a = b, then a + c = b + c

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 .

White Board Practice


VW = XY implies VW XY


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


K is the midpoint of JL only if JK = KL


If a number is divisible by 4, then it is divisible by 6.


n > 8 only if n is greater than 7


I’ll dive if you dive


If x2 = 49, then x = 7.


r + n = s + n if r = s


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

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

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.

Example 1Given : m 1 = m 2;

AD bisects CAB;

BD bisects CBA

Prove: m 3 = m 4

4A

213

B

C

D

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 Reasons1. 1.2. 2.3. 3.4. 4.Etc… Etc…

Example 1Given : m 1 = m 2;

AD bisects CAB;

BD bisects CBA

Prove: m 3 = m 4

4A

213

B

C

D

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

Try itGiven : WX = YZ

Y is the midpoint of XZ

Prove: WX = XY

W ZYX

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

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

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.


• Two right angles are ____________ complementary.


• Right angles are ___________ vertical angles.


• Angles A, B, and C are __________ complementary.


• Vertical angles ___________ have a common supplement.


• Find the measure of a complement and a supplement of T.

m T = 40



m T = 89



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

http://www.viewimages.com/AltaVistaView.asp?imageid=171075&promotionid=1&partnerid=2&type=results

http://www.viewimages.com/AltaVistaView.asp?imageid=171075&promotionid=1&partnerid=2&type=results

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

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

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

Practice

• Given: 2 and 3 are supplementary

Prove: m 1 = m 3

1 24

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

Practice

• Given: m 1 = m 4

Prove: 4 is supplementary to 2

1 24

3

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

Chapter 2 Deductive Reasoning Learn deductive logic Do your first 2- column proof New Theorems and...

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Transcript of Chapter 2 Deductive Reasoning Learn deductive logic Do your first 2- column proof New Theorems and...