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