chapter 2 deductive reasoning

Chapter 2Deductive Reasoning

• Learn deductive logic• Do your first 2-

column proof• New Theorems and

Postulates

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.

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.

Which is the hypothesis?

Which is the conclusion?

You are my student

you love Geometry

If you are my student, then you love Geometry.

The Converse

A conditional with the hypothesis and conclusion reversed.

If q, then p.

hypothesis conclusion

If it is daytime, then the sun is shining.

Original: If the sun is shining, then it is daytime.

The Counterexample

If p, then q

TRUEFALSE

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

The Counterexample

• This is HARD !

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

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

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

If m1 = 90, then 1 is a right angle.

If 1 is a right angle, then m1 = 90.

m1 = 90 if and only if 1 is a right angle.

m1 = 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

if x = y, then either x or y may be substituted for the other in any equation.

Reflexive

Property

x = x.

A number equals itself.

Symmetric Property

if x = y, then y = x.

Order of equality does not matter.

Transitive

Property

if x = y and y = z, then x = z.

Two numbers equal to the same number are equal to each other.

Properties of CongruenceReflexive

Property

AB AB≅

A segment (or angle) is congruent to itself

Symmetric Property

If AB CD, then CD AB≅ ≅

Order of equality does not matter.

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.

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. =

STATEMENTS REASONS

Your Second Proof

Given: AB = CDProve: AC = BD

1. AB = CD 1. Given

2. AB + BC = BC + CD 2. Addition Prop. =

3. AB + BC = AC 3. Segment Addition Post.

BC + CD = BD

4. AC = BD 4. 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

Important Notes

• Does the order matter?

• Don’t leave out steps Don’t ASSume

Statements Reasons

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 BXC 2. Definition of Angle Bisector or ABX m BXC3.m ABX + m BXC = 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 Equality6. m BXC = 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

• 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 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

2.4 Special Pairs of Angles

Objectives

• Apply the definitions of complimentary and supplementary angles

• State and apply the theorem about vertical angles

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

See It!

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

See It!

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

Vertical AnglesTwo angles formed on the opposite sides of

the intersection of two lines.

1

2

3

4

TheoremVertical angles are congruent

1

2

3

4

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

angles are right.l

m

See It!

TheoremIf two lines are perpendicular, then they form

congruent, adjacent angles.

l

m1 2

If l m, then

1 2.

TheoremIf two lines intersect to form congruent, adjacent

angles, then the lines are perpendicular.

l

m1 2

If 1 2, then

l m.

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.

See It!

Construction 4Given a segment, construct the perpendicular bisector of the segment.

Given:

Construct:

Steps:

ABbisector of AB

Construction 5Given 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 6Given 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

TheoremIf 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

TheoremIf 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

Practice

• Given: m 1 = m 4

Prove: 4 is supplementary to 2

1 24

3