pre-ap bellwork 7) the radius of a circle is 4 feet. describe what happens to the circle’s area...
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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.
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Pre-AP Bellwork
8) Use the letters of the alphabet and create two different sequences that begin with the same two letters.
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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.
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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.
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Reasoning and ProofChapter 2
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2-1 Conditional Statements
What is a conditional statement?
How do you write the converse of a conditional statement?
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2-1Conditional Statements
Conditional
An if – then statement
Two Parts:
Hypothesis – The part following the if Conclusion – The part following the then
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2-1 Conditional Statements
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2-1 Conditional Statements
If today is the first day of fall,
then the month is September. Hypothesis:
Conclusion:
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2-1 Conditional Statements
If y – 3 = 5,
then y = 8. Hypothesis:
Conclusion:
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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?
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2-1Conditional Statements
A tiger is an animal.
If something is a tiger, then it is an animal.
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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.
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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.
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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.
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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.
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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
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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
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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.
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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.
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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.
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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.
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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.
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2-1 Conditional Statements
Homework Pages 72 – 73 33 – 39; 42; 43; 47
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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
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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.
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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
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5-4 Inverses, Contrapositives, and Indirect Reasoning
Contrapositive
Switches the hypothesis and conclusion
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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.
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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..