CHAPTER 2nolongerinuse2015.weebly.com/uploads/9/9/0/3/9903287/chapter_2.pdf2-2 Conditional Statements •What are the converse, inverse, and contrapositive of the conditional statements

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.


• Pattern #1:

• 3, 9, 27, 81,…

• Pattern #2:

• Pattern #3:


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

• 3, 9, 27, 81, 243, 729,…

• Pattern #2:


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


• 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.



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


• 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.


• 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


• 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.


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

• Vertical angles share a vertex.

• Dolphins are mammals.


• 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.


• 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.




• 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.


• 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.


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


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)


• 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)


• 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


• Biconditional

• “…if and only if…”

• p↔q

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

• p→q and q → p


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


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. An integer less than zero is negative.


2. An integer less than zero is negative.

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


3. P: an animal is an amphibian

Q: an animal can live on land and in water


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.


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


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.


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


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


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



• True or false?



• True

• What is the converse?



• True

• What is the converse?

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



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

• Is the converse true?




• 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.


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


• Conditional: p → q

• Converse: q → p

• Biconditional: p ↔ q


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

• If true, then write the converse

• If false, give a counterexample



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

• If true, then write the biconditional

• If false, give a counterexample



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.


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.


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.


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.


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

• Is this statement reversible?

• If yes, then write it as a true biconditional


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


A good definition:

• Is reversible (can be written as a biconditional)

• Uses clearly defined terms (specifics!)

• Is precise


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.


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.


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.


• 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.


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,


• 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.


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

• Angle 1 and 2 share a common vertex.


• 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)


Law of Syllogism

• If pq is true

• And qr is true,

• Then p r is true.

This makes a valid conclusion.


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.


• 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.


• 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.


• 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











• 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.



2-6 Proving Angles Congruent

• See online text – interactive path

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CHAPTER 2nolongerinuse2015.weebly.com/uploads/9/9/0/3/9903287/chapter_2.pdf2-2 Conditional Statements •What are the converse, inverse, and contrapositive of the conditional statements