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CHAPTER 2 Reasoning and Proof

2-1 Patterns and Inductive Reasoning

• Inductive Reasoning- is reasoning based on patterns you observe.

• Conjecture- a conclusion you reach using inductive reasoning.

• Counterexample- an example that shows that a conjecture is incorrect.

2-1 Patterns and Inductive Reasoning

• Pattern #1:

• 3, 9, 27, 81,…

• Pattern #2:

• Pattern #3:

2-1 Patterns and Inductive Reasoning

• Pattern #1: (Each term is 3x the previous term)

• 3, 9, 27, 81, 243, 729,…

• Pattern #2:

2-1 Patterns and Inductive Reasoning

• What conjecture can you make about the number of regions 20 diameters form?

2-1 Patterns and Inductive Reasoning

• What conjecture can you make about the sum of the first 30 even numbers?

2-1 Patterns and Inductive Reasoning

• What conjecture can you make about the sum of the first 30 even numbers?

• First, find the first few sums and look for a pattern.

2-1 Patterns and Inductive Reasoning

• What conjecture can you make about the sum of the first 30 even numbers?

• First, find the first few sums and look for a pattern.

2-1 Patterns and Inductive Reasoning

• Find a counterexample to each of the following conjectures

1. If the name of a month starts with the letter J, it is a winter month.

2. You can connect any three points to form a triangle.

3. When you multiply a number by 2, the product is greater than the original number.

2-1 Patterns and Inductive Reasoning

• Find a counterexample to each of the following conjectures

4. If a flower is red, it is a rose.

5. One and only one plane exists through any three points.

6. When you multiply a number by 2, the product is divisible by 6.

2-2 Conditional Statements

2-2 Conditional Statements

• What is hypothesis and the conclusion of the conditional statement:

• If an animal is a robin, then the animal is a bird.

• If an angle measures 130, then the angle is obtuse.

2-2 Conditional Statements

• How can we write the following statements as a conditional statements?

• Vertical angles share a vertex.

• Dolphins are mammals.

2-2 Conditional Statements

• Truth Value- of every conditional statement is either true (1) or false (0). • In order to show that a conditional

statement is true, we must show that every time the hypothesis is true, the conclusion is also true.

• In order to show that a conditional is false, we must find one counterexample for which the hypothesis is true and the conclusion is false.

2-2 Conditional Statements

• Negation of a statement p is the opposite of the statement.

• ¬𝑝 or ∼ 𝑝 (both represent “not p”)

• What are the negations of the following statements?

• The sky is blue.

2-2 Conditional Statements

2-2 Conditional Statements

2-2 Conditional Statements

• What are the converse, inverse, and contrapositive of the conditional statements below?

1. If the figure is a square, then the figure is a quadrilateral.

2. If a vegetable is a carrot, then it contains beta carotene.

2-2 Conditional Statements

• What are the converse, inverse, and contrapositive of the conditional statements below? 1. If the figure is a square, then the figure

is a quadrilateral. First, define hypothesis and conclusion, p

and q.

Then write the p and q statements for the converse, inverse, and contrapositive.

Then determine if each statement is true or false, 1 or 0.

2-2 Conditional Statements

𝑝: 𝑇ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒 𝑖𝑠 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑞: 𝑇ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒 𝑖𝑠 𝑎 𝑞𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙

Conditional: 𝒑 → 𝒒 If the figure is a square, then the figure is a

quadrilateral

Converse: 𝒒 → 𝒑 If the figure is a quadrilateral, then the figure is a

square

Inverse: ¬𝒑 → ¬𝒒 If the figure is not a square, then the figure is not

quadrilateral

Contrapositive: ¬𝒒 → ¬𝒑 If the figure is not quadrilateral, then the figure is

not a square

2-2 Conditional Statements

Conditional: 𝒑 → 𝒒 If the figure is a square, then the figure is a

quadrilateral Converse: 𝒒 → 𝒑 If the figure is a quadrilateral, then the figure is a

square False(0) Counterexample: a rectangle that isn’t a

square

Inverse: ¬𝒑 → ¬𝒒 If the figure is not a square, then the figure is not

quadrilateral False(0)Counterexample: a kit has four sides and is

not square

Contrapositive: ¬𝒒 → ¬𝒑 If the figure is not quadrilateral, then the figure is not a

square True (1)

2-2 Conditional Statements

• A conjunction (and) is only true (1) when both s and j are true (1) • A disjuction (or) is only false (0) when both s and j are false (0)

2-2 Conditional Statements

• Truth table- lists all the possible combinations of truth values for two or more statements.

2-3 Biconditionals and Definitions

• Biconditional - a single true statement that combines a true conditional and its true converse.

• Good definitions can be written as biconditionals

2-3 Biconditionals and Definitions

• Biconditional

• “…if and only if…”

• p↔q

• Biconditionals are combinations of a conditional and it’s converse:

• p→q and q → p

2-3 Biconditionals and Definitions

1. A pentagon is a polygon that has five sides.

2-3 Biconditionals and Definitions

1. A pentagon is a polygon that has five sides.

1. A pentagon is a polygon if and only if it has five sides.

2-3 Biconditionals and Definitions

2. An integer less than zero is negative.

2-3 Biconditionals and Definitions

2. An integer less than zero is negative.

2. An integer is negative if and only if it is less than zero.

2-3 Biconditionals and Definitions

3. P: an animal is an amphibian

Q: an animal can live on land and in water

2-3 Biconditionals and Definitions

3. P: an animal is an amphibian

Q: an animal can live on land and in water

3. An animal is an amphibian if and only if an animal can live on land and in water.

2-3 Biconditionals and Definitions

4. Two lines are parallel if and only if they do not intersect.

2-3 Biconditionals and Definitions

4. Two lines are parallel if and only if they do not intersect.

• P→Q: if two lines are parallel then they do not intersect.

• Q→P: if two lines do not intersect then they are parallel.

2-3 Biconditionals and Definitions

5. A number is even if and only if it is divisible by 2

2-3 Biconditionals and Definitions

5. A number is even if and only if it is divisible by 2

• P→Q: If a number is even then it is divisible by 2

• Q→P: if a number is divisible by 2 then it is an even number

2-3 Biconditionals and Definitions

Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.

2-3 Biconditionals and Definitions

Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.

• True or false?

2-3 Biconditionals and Definitions

Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.

• True

• What is the converse?

2-3 Biconditionals and Definitions

Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.

• True

• What is the converse?

• If two angles are supplementary then the sum of their measures is 180.

2-3 Biconditionals and Definitions

Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.

Converse: If two angles are supplementary then the sum of their measures is 180.

• Is the converse true?

2-3 Biconditionals and Definitions

Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary.

Converse: If two angles are supplementary then the sum of their measures is 180.

• True

• Rewrite as a biconditional.

2-3 Biconditionals and Definitions Conditional: If the sum of the measures

of two angles is 180, then the two angles are supplementary.

Converse: If two angles are supplementary then the sum of their measures is 180.

Biconditional: Two angles are supplementary if and only if the sum of the measures of the two angles is 180

2-3 Biconditionals and Definitions

• Conditional: p → q

• Converse: q → p

• Biconditional: p ↔ q

2-3 Biconditionals and Definitions

Conditional: If two angles have equal measures, then the angles are congruent.

• If true, then write the converse

• If false, give a counterexample

2-3 Biconditionals and Definitions

Conditional: If two angles have equal measures, then the angles are congruent.

Converse: If angles are congruent then the two angles have equal measures.

• If true, then write the biconditional

• If false, give a counterexample

2-3 Biconditionals and Definitions

Conditional: If two angles have equal measures, then the angles are congruent.

Converse: If angles are congruent then the two angles have equal measures.

Biconditional: Two angles have equal measures if and only if they are congruent.

2-3 Biconditionals and Definitions

Biconditional: Two angles have equal measures if and only if they are congruent.

• What are the two conditional statements that form this biconditional?

2-3 Biconditionals and Definitions Biconditional: Two angles have equal

measures if and only if they are congruent.

• What are the two conditional statements that form this biconditional? • If two angles have equal measures then

they are congruent.

• If two angles are congruent then they have equal measures.

2-3 Biconditionals and Definitions

Biconditional: Two numbers are reciprocals if and only if their product is 1.

• What are the two conditional statements that form this biconditional?

2-3 Biconditionals and Definitions Biconditional: Two numbers are

reciprocals if and only if their product is 1.

• What are the two conditional statements that form this biconditional? • If two numbers are reciprocals then

their product is 1.

• If the product of two numbers is 1 then they are conditional.

2-3 Biconditionals and Definitions

Definition: A quadrilateral is a polygon with four sides.

• Is this statement reversible?

• If yes, then write it as a true biconditional

2-3 Biconditionals and Definitions Definition: A quadrilateral is a polygon

with four sides.

• Conditional: if a figure is a quadrilateral, then it is a polygon with four sides.

• Converse: if a figure is a polygon with four sides, then it is a quadrilateral.

• Biconditional: A figure is a quadrilateral if and only if it is a polygon with four sides.

2-3 Biconditionals and Definitions

Definition: A straight angle is an angle that measures 180.

• Is this statement reversible?

• If yes, then write it as a true biconditional

2-3 Biconditionals and Definitions

Definition: A straight angle is an angle that measures 180.

Conditional: if an angle is straight then it measures 180

Converse: if an angle measures 180 then it is straight

Biconditional: an angle is straight if and only if it measures 180

2-3 Biconditionals and Definitions

A good definition:

• Is reversible (can be written as a biconditional)

• Uses clearly defined terms (specifics!)

• Is precise

2-3 Biconditionals and Definitions

A square is a figure with four right angles.

• Good definition? Or not? Why?

2-4 Deductive Reasoning

Deductive (or logical) reasoning: the process of reasoning logically from given statements or facts to a conclusion.

2-4 Deductive Reasoning

Law of Detachment

• If the hypothesis (p) of a true conditional is true, then the conclusion is true also.

• If pq is true

• And p is true,

• Then q is true.

This makes a valid conclusion.

2-4 Deductive Reasoning

What conclusion can you make from this true statement?

• If a student gets an A on a final exam, then the student will pass the course.

• Felicia got an A on her history final exam.

2-4 Deductive Reasoning

• If a student gets an A on a final exam, then the student will pass the course.

• Felicia got an A on her history final exam.

• Therefore, by the law of detachment, Felicia will pass her history class/course.

2-4 Deductive Reasoning

What conclusion can you make from this true statement?

• If a ray divides an angle into two congruent angles, then the ray is an angle bisector.

• And,

2-4 Deductive Reasoning

• If a ray divides an angle into two congruent angles, then the ray is an angle bisector.

• And,

• Therefore, by the law of detachment, ray AB is an angle bisector.

2-4 Deductive Reasoning

• If two angles are adjacent, then they share a common vertex.

• Angle 1 and 2 share a common vertex.

2-4 Deductive Reasoning

• If two angles are adjacent, then they share a common vertex.

• Angle 1 and 2 share a common vertex.

• We can NOT make a conclusion because the statement does not match the hypothesis (it matches the conclusion)

2-4 Deductive Reasoning

Law of Syllogism

• If pq is true

• And qr is true,

• Then p r is true.

This makes a valid conclusion.

2-4 Deductive Reasoning

Law of Syllogism • If pq is true • And pr is true, • Then p r is true.

• Example: • If it is July, then you are on summer

break. • If you are on summer break, then you

work at a smoothie shop. • You CONCLUDE: If it is July, then you

work at a smoothie shop.

2-4 Deductive Reasoning

• If a figure is a square, then the figure is a rectangle. If a figure is a rectangle, then the figure has four sides.

• Therefore, by the law of syllogism, if a figure is a square, then the figure has four sides.

2-4 Deductive Reasoning

• If you do gymnastics, then you are flexible. If you do ballet, then you are flexible.

• Both statements have the same conclusion, so we cannot use the law of syllogism and cannot make a conclusion.

2-4 Deductive Reasoning

• If you are in Mrs. Lawson’s Geometry class, then you attend GMS. If you attend GMS, then you currently live in Florida.

• Sophia is in Mrs. Lawson’s Geometry class. • Therefore, by the law of syllogism, if

you are in Mrs. Lawson’s class then you currently live in Florida.

• And, by the law of detachment, Sophia lives in Florida.

2-5 Reasoning in Algebra and Geometry

• Addition Property

• Subtraction Property

• Multiplication Property

• Division Property

• Reflexive Property

• Symmetric Property

• Transitive Property

• Substitution Property

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

• A proof is a convincing argument that uses deductive reasoning.

• It logically shows why a conjecture is true.

• A two-column proof lists each statement on the left and gives the justification on the right.

2-5 Reasoning in Algebra and Geometry

2-5 Reasoning in Algebra and Geometry

2-6 Proving Angles Congruent

• See online text – interactive path