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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. 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.
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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.
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.
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 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. Illinois Chicago
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 • 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.
