Pre-AP Bellwork

7) The radius of a circle is 4 feet. Describe what happens to the circle’s area when the radius is doubled.

Pre-AP Bellwork

8) Use the letters of the alphabet and create two different sequences that begin with the same two letters.

Pre-AP Bellwork

9) Draw a Venn Diagram to illustrate the following conditional statement.

If the game is baseball, then the game is a team sport.

Pre-AP Bellwork

10) Write the sentence as a conditional statement: Two complementary angles form a right angle.

Write the converse, inverse, and contrapositive of the conditional.

Reasoning and ProofChapter 2

2-1 Conditional Statements

What is a conditional statement?

How do you write the converse of a conditional statement?

2-1Conditional Statements

Conditional

An if – then statement

Two Parts:

Hypothesis – The part following the if Conclusion – The part following the then

2-1 Conditional Statements

2-1 Conditional Statements

If today is the first day of fall,

then the month is September. Hypothesis:

Conclusion:

2-1 Conditional Statements

If y – 3 = 5,

then y = 8. Hypothesis:

Conclusion:

2-1 Conditional Statements

Many sentences can be written as conditionals.

Can you identify the hypothesis and conclusion?

Did you know a rectangle has four

right angles?

So, you are saying that if a figure is a rectangle, then it

has four right angles?

2-1Conditional Statements

A tiger is an animal.

If something is a tiger, then it is an animal.

2-1 Conditional Statements

Write each sentence as a conditional. An integer that ends with 0 is divisible by 5.

A square has four congruent sides.

If an integer ends with 0, then it is divisible by 5.

If a figure is a square, then it has 4 congruent sides.

2-1 Conditional Statements

Truth Value

True or False

A conditional is proven true if every time the hypothesis is true, the conclusion is also true.

A conditional only needs 1 counterexample to be proven false.

2-1 Conditional Statements

Show the conditional is false by finding a counterexample: If it is February, then there are only 28 days in the

month.

Since 2008 was a leap year, February had 29 days.

2-1 Conditional Statements

Show the conditional is false by finding a counterexample: If the name of a state contains the word New,

then the state borders an ocean.

New Mexico is a state, but it does not border an ocean.

2-1 Conditional Statement

A Venn diagram can be used to better understand true conditional statements.

If you live in Chicago, then you live in Illinois.

Chicago

Illinois

2-1 Conditional Statements

Draw a Venn diagram to illustrate this conditional:

If something is a cocker spaniel, then it is a dog.

Dog

Cocker Spaniel

2-1 Conditional Statements

Converse

Switches the hypothesis and conclusion of a conditional.

Conditional: If two lines intersect to form right angles, then they are perpendicular.

Converse: If two lines are perpendicular, then they intersect to form right angles.

2-1 Conditional Statements

Write the converse of the following conditional.

Conditional If two lines are not parallel and do not intersect, then they

are skew.

Converse If two lines are skew, then they are not parallel and do not

intersect.

2-1 Conditional Statements

In the last two examples, both the conditional and its converse are true.

This is not always the case.

Conditional: If a figure is a square, then it has 4 sides.

Converse: If a figure has 4 sides, then it is a square.This is not true, as any rectangle can be used as a counterexample.

2-1 Conditional Statements

Write the converse of each conditional statement. Determine the truth value of the conditional and its converse. If two lines do not intersect, then they are parallel.

If two lines are parallel, then they do not intersect.

The conditional is false, but the converse is true. If x = 2, then |x| = 2.

If |x| = 2, then x = 2.

The conditional is true, but the converse is false.

2-1 Conditional Statements

Conditional Statements and Converses

Statement Example Symbolic Form

You Read It

Conditional If an angle is a straight angle,

then its measure is 180.

p → q If p, then q.

Converse If the measure of an angle is 180,

then it is a straight angle.

q → p If q, then p.

2-1 Conditional Statements

Homework Pages 72 – 73 33 – 39; 42; 43; 47

5-4 Inverses, Contrapositives, and Indirect Reasoning

Negation

Opposite truth value

“Knoxville is the capital of Tennessee.” False

Negation: “Knoxville is not the capital of Tennessee.” True

5-4 Inverses, Contrapositives, and Indirect Reasoning

Write the negation for each statement.

Angle ABC is obtuse. Angle ABC is not obtuse.

Lines m and n are not perpendicular. Lines m and n are perpendicular.

5-4 Inverses, Contrapositives, and Indirect Reasoning

Inverse

Negates the hypothesis and conclusion of a conditional statement.

Conditional If a figure is a square, then it is a rectangle.

Inverse If a figure is not a square, then it is not a rectangle

5-4 Inverses, Contrapositives, and Indirect Reasoning

Contrapositive

Switches the hypothesis and conclusion

5-4 Inverses, Contrapositives, and Indirect Reasoning

Conditional If a figure is a square, then it is a rectangle.

Inverse If a figure is not a square, then it is not a rectangle.

Contrapositive If a figure is not a rectangle, then it is not a square.

Conditional Statements and ConversesStatement Example Symbolic

FormYou Read It

Conditional If an angle is a straight angle,

then its measure is 180.

p → q If p, then q.

Converse If the measure of an angle is 180,

then it is a straight angle.

q → p If q, then p.

Negation An angle is not a straight angle.

~p Not p.

Inverse If an angle is not a straight angle,

then its measure is not 180.

~p → ~q If not p, then not q.

Contrapositive If an angle’s measure is not

180, then it is not a straight angle.

~q → ~p If not q, then not p..