PreAP Bellwork 7 The radius of a circle
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 Proof Chapter 2
2 -1 Conditional Statements l What is a conditional statement? l How do you write the converse of a conditional statement?
2 -1 Conditional Statements l Conditional l An if – then statement l Two Parts: l l Hypothesis – The part following the if Conclusion – The part following 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 l Many sentences can be written as conditionals. 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? Can you identify the hypothesis and conclusion?
2 -1 Conditional Statements A tiger is an animal. If something is a tiger, then it is an animal.
2 -1 Conditional Statements l Write each sentence as a conditional. l An integer that ends with 0 is divisible by 5. l A square has four congruent sides. l If an integer ends with 0, then it is divisible by 5. l If a figure is a square, then it has 4 congruent sides.
2 -1 Conditional Statements l Truth Value l True or False l A conditional is proven true if every time the hypothesis is true, the conclusion is also true. l A conditional only needs 1 counterexample to be proven false.
2 -1 Conditional Statements l Show the conditional is false by finding a counterexample: l If it is February, then there are only 28 days in the month. l Since 2008 was a leap year, February had 29 days.
2 -1 Conditional Statements l Show the conditional is false by finding a counterexample: l If the name of a state contains the word New, then the state borders an ocean. l New Mexico is a state, but it does not border an ocean.
2 -1 Conditional Statement l A Venn diagram can be used to better understand true conditional statements. l If you live in Chicago, then you live in Illinois Chicago
2 -1 Conditional Statements l Draw a Venn diagram to illustrate this conditional: l If something is a cocker spaniel, then it is a dog. Dog Cocker Spaniel
2 -1 Conditional Statements l Converse l 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 l Write the converse of the following conditional. Conditional l If two lines are not parallel and do not intersect, then they are skew. Converse l If two lines are skew, then they are not parallel and do not intersect.
2 -1 Conditional Statements l In the last two examples, both the conditional and its converse are true. l 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 l Write the converse of each conditional statement. Determine the truth value of the conditional and its converse. l l 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 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 l Homework l l Pages 72 – 73 33 – 39; 42; 43; 47
5 -4 Inverses, Contrapositives, and Indirect Reasoning l Negation l Opposite truth value l “Knoxville is the capital of Tennessee. ” l l False Negation: “Knoxville is not the capital of Tennessee. ” l True
5 -4 Inverses, Contrapositives, and Indirect Reasoning l Write the negation for each statement. l l 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 l Inverse l Negates the hypothesis and conclusion of a conditional statement. l Conditional l l If a figure is a square, then it is a rectangle. Inverse l If a figure is not a square, then it is not a rectangle
5 -4 Inverses, Contrapositives, and Indirect Reasoning l Contrapositive l Switches the hypothesis and conclusion
5 -4 Inverses, Contrapositives, and Indirect Reasoning l Conditional l l Inverse l l If a figure is a square, then it is a rectangle. If a figure is not a square, then it is not a rectangle. Contrapositive l If a figure is not a rectangle, then it is not a square.
Conditional Statements and Converses Statement Example Symbolic You Read It Form p→q If p, then q. Conditional If an angle is a straight angle, then its measure is 180. 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. .
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