Inductive Reasoning

• Observation (Given)

• Conjecture(s) (educated guess)

• True Example(s)

• One Counterexample

Writing Conjectures

• She refused your request for a date• WMD were NOT found in Iraq• No terrorist acts in U.S. for last 3 years• Mars rover found rounded, smooth stones• The sun is near the eastern horizon• The moon is brightly visible• The grass is wet• The bread popped up, but was not toasted• The car won’t start

Given:

Writing Conjectures

• Someone has more than 3 exceptions in this class

• Joe scored a level 5 on his FCAT math exam• Molly is a cheerleader• Gary is absent often• Billy is a goat• Henry is sleeping in art class• Max has a positive, “can do” attitude• I imagine 3 noncollinear points• The polygon has 7 sides

Given:

• Given: Points A, B, C are collinear

– Conjecture 1: B is between A and C

– Conjecture 2: Only 1 plane can be constructed

– Conjecture 3: Exactly 1 line with point B can be constructed

Are the conjectures true?Find one counterexample

Are the conjectures true?

• Given: 2 intersecting lines

– Conjecture 1: Exactly 3 angles are formed

– Conjecture 2: Adjacent angles are linear pairs

– Conjecture 3: Exactly 1 pair of vertical angles are formed

Find one counterexample

• Given: a perimeter of 80 feet– Conjecture 1: A rectangle of length 25 and

width 20 can be constructed

– Conjecture 2: 2 squares with different side lengths can be constructed.

– Conjecture 3: The largest quadrilateral area that can be enclosed is 25 X 15 = 375 sq ft

Are the conjectures true?Find one counterexample

Inductive Reasoning

• Given a fact

• State/write a conjecture (educated guess)

• Find true examples

• Find One Counterexample

• Given: 3 noncollinear point

– Conjecture 1: Exactly one line can be drawn

– Conjecture 2: 2 Exactly one plane can be drawn

Are the conjectures true?Find one counterexample

Conjunctions and Disjunctions

• It’s raining It’s Monday

• It’s raining and It’s Monday

• It’s raining or It’s Monday

Λ

ν

Negations

• It’s raining It’s Monday

• It’s NOT raining and It’s Monday

• It’s raining or It’s NOT Monday

Λ

ν ~

~

Paper50

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Communities that recycle

VENN DIAGRAM

p q p Λ q p ν q

T T T T

T F F T

F T F T

F F F F

TRUTH TABLE

• It’s raining It’s Monday

p q p Λ q p ν q

T T T T

T F F T

F T F T

F F F F

TRUTH TABLE• I broke curfew I’m grounded

TRUTH TABLE• It’s raining It’s Monday It’s 3rd period

p q r p Λ q (p Λ q) ν r

T T T T T

T T F T T

T F T F T

T F F F F

F T T F T

F T F F F

F F T F T

F F F F F

Conditionals• If it rained, then the grass is wet.

• If there was life on Mars, then there was water on Mars.

• If it’s a duck, then it quacks.

• If 2 angles are supplementary, then their sum is 180 degrees.

• If a polygon is a triangle, then it has 3

sides.

Write Conditionals

• Ducks Quack

• She only dates handsome men

• It’s dark during the night

• 2 Perpendicular lines form 4 right angles

• Linear pairs are supplementary

• 3 noncollinear points determine a plane

• Vertical angles are congruent

Conditional

• If it rained, then the grass is wet.

• If– It rained

• Then– The grass is wet

Converse

• If it’s a duck, then it flies.

• If it flies, then it’s a duck

Inverse

• If it’s a duck, then it flies.

• If it’s NOT a duck, then it does NOT fly.

Contrapositive

• If it’s a duck, then it flies.

• If it does NOT fly, then it’s NOT a duck

Converse

• If it rained, then the grass is wet.

• If the grass is wet, then it rained

Inverse

• If it rained, then the grass is wet.

• If it did NOT rain, then the grass is NOT wet

Contrapositive

• If it rained, then the grass is wet.

• If the grass is NOT wet, then it did NOT rain.

p q

p Λ q

p ν q p

q

Write Conditionals

• Given: Ducks are birds

• Write the conditional:

• Write the converse:

• Write the inverse:

• Write the contrapositive:

Law of Detachment

• 1. If it’s a duck then it flies

• 2. It’s a duck

• 3. CONCLUSION: it flies

Law of Detachment

• 1. If it’s a duck then it flies

• 2. It flies

• 3. CONCLUSION: it’s a duck

Law of Detachment• 1. If then

• 2.

• 3. CONCLUSION:

Law of Detachment• 1. If it rained, then the grass is wet.• 2. It rained• 3. CONCLUSION: the grass is wet

Law of Detachment• 1. If then

• 2.

• 3. CONCLUSION:

Law of Detachment

• 1. If it rains, then the grass will get wet.

• 2. The grass is wet

• 3. CONCLUSION: it rained

Law of Syllogism• 1. If it rains, then the grass is wet

• 2. If the grass is wet, then I won’t mow.

• 3. It rained

• 4. CONCLUSION: I won’t mow

Law of Syllogism• 1. If it rains, then the grass is wet

• 2. If the grass is wet, then I won’t mow.

• 3. I didn’t mow

• 4. CONCLUSION: The grass is wet

Law of Syllogism• 1. If it’s a duck, then it flies.

• 2. If it flies, then it has wings.

• 3. It’s a duck

• 4. CONCLUSION: it has wings

Law of Syllogism• 1. If it’s a duck, then it flies.

• 2. If it flies, then it has wings.

• 3. It has wings

• 4. CONCLUSION: It’ a duck

Law of Syllogism• 1. If it rains, then the grass will get wet.

• 2. If the grass is wet, I won’t mow

• 3. It rained

• 4. CONCLUSION: I won’t mow

Deductive Reasoning

• 1. If Alex takes the car to the store, he will stop at the post office.

• 2. If Alex stops at the post office, he will buy stamps.

What can you conclude using Law of Syllogism?

Deductive Reasoning

If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman.

a. The circus is in town

b. There are tents at the fairgrounds

Write your conclusions:

1. If a. above is true

2. If b. above is true

Postulates

1. Thru any 2 points, there is exactly one line

2. Thru any 3 noncollinear points, there is exactly one plane

3. A line contains at least 2 points

4. A plane contains at least 3 noncollinear points

Postulates

5. If 2 points lie in a plane, then the line containing those points lies in the plane

6. If 2 lines intersect, then their intersection is exactly one point

7. If 2 planes intersect, then their intersection is a line

Properties of Equality1. Reflexive: a = a

2. Symmetric: If a = b, then b = a

3. Transitive: If a = b, and b = c, then a = c

4. Addition & Subtraction: If a = b, then a + c = b + c. If a = b , then a – c = b – c

5. Multiplication & Division: If a = b, then ac = bc. If a = b, then a/c = b/c (c 0)

6. Substitution: If a = b, then a may be replaced with b

7. Distributive: a(b + c) = ab + ac

Name the property

1 If 3x = 120, then x = 40

2 If 13 = AB, then AB = 13

3 If y = 75, and y = mA, then mA = 75

4 If AB = BC, and BC = CD, then AB = CD

2 Column Proof

Given:

3x + 5

2

Prove:

x = 3

= 7

Statements Reasons

Properties of Segments

1. Reflexive: AB = AB

2. Symmetric: If AB = CD, then CD = AB

3. Transitive: If AB = CD, and CD = EF, then AB = EF

Properties of Angles

1. Reflexive: m1 = m1

2. Symmetric: If m1 = m2, then m2 = m1

3. Transitive: If m1 = m2, and m2 = m3, then m1 = m3

Segment Postulates

1. Ruler Postulate: Any segment can be measured

2. Segment Addition Postulate: If B is between A and C, then AB + BC = AC

Properties of Segment Congruence

1. Reflexive: AB AB

2. Symmetric: If AB CD, then CD AB

3. Transitive: If AB CD, and CD EF, then AB EF

Angle Postulates

1. Protractor Postulate: Any angle can be measured

2. Angle Addition Postulate: If R is inside PQS, then mPQR + mRQS = mPQS

Properties of Angles

1. Reflexive: m1 m1

2. Symmetric: If m1 m2, then m2 m1

3. Transitive: If m1 m2, and m2 m3, then m1 m3

Angle Theorems1. Linear pairs are supplementary

2. Adjacent angles that form a right angle are complementary

3. Angles supplementary to the same angle or to congruent angles are congruent.

4. Angles complementary to the same angle or to congruent angles are congruent

5. Vertical angles are congruent

Angle Theorems6. Perpendicular lines intersect to form 4

right angles

7. All right angles are congruent

8. Perpendicular lines form congruent adjacent angles

9. If 2 angles are congruent and supplementary, then each angle is a right angle

10. If 2 congruent angles form a linear pair, then they are right angles

Name the property1 m1 = m1

2 If AB + BC = DE + BC, then AB = DE

3 If XY = PQ and XY = RS, then PQ = RS

4 If ⅓ x = 5, then x = 15

5 If 2x = 9, then x = 9/2

2 Column Proof

Given:

PR = QS

Prove:

PQ = RS

Statements Reasons

PQ

R

S