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1 Click the mouse button or press the Space Bar to display the answers.
2
3 Questions On yesterday s Assignment?
4 2-3 Objectives You will learn to: Write the converse, inverse, and contrapositive of if-then statements.
5 Vocabulary Conditional Statement If-Then Statement Hypothesis Conclusion Related Conditionals Converse Inverse Contrapositive Logically Equivalent
6 Truth Values A conditional statement has a truth value of either true (T) or false (F). It is false only when the hypothesis (p) is true and the conclusion (q) is false. To show that a conditional statement is false, you need to find only one counterexample where the hypothesis is true and the conclusion is false. p q p q T T F F T F T F T F T T
7 Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 10 days, and he still has a hurt ankle. The hypothesis is true, but the conclusion is false. Answer: Since the result is not what was expected, the conditional statement is false.
8 Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 3 days, and he still has a hurt ankle. The hypothesis is false, and the conclusion is false. The statement does not say what happens if Yukon only rests for 3 days. His ankle could possibly still heal. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true.
9 Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 10 days, and he does not have a hurt ankle anymore. The hypothesis is true since Yukon rested for 10 days, and the conclusion is true because he does not have a hurt ankle. Answer: Since what was stated is true, the conditional statement is true.
10 Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 7 days, and he does not have a hurt ankle anymore. The hypothesis is false, and the conclusion is true. The statement does not say what happens if Yukon only rests for 7 days. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true.
11 Determine the truth value of the following statements for each set of conditions. If it rains today, then Michael will not go skiing. a. It does not rain today; Michael does not go skiing. Answer: true b. It rains today; Michael does not go skiing. Answer: true c. It snows today; Michael does not go skiing. Answer: true d. It rains today; Michael goes skiing. Answer: false
12 Key Concepts Statement Formed By Symbol Examples Conditional Converse Inverse Given hypothesis and conclusion Exchanging the hypothesis and conclusion of the conditional Negating both the hypothesis and conclusion of the conditional p q q p ~p ~q If two angles have the same measure, then they are congruent. If two angles are congruent, then they have the same measure. If two angles do not have the same measure, then they are not congruent. Contrapositive Negating both the hypothesis and conclusion of the converse statement ~q ~p If two angles are not congruent, then they do not have the same measure.
13 Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. First, write the conditional in if-then form. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. The converse is false. A rectangle with = 2 and w = 4 is not a square.
14 Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. Inverse: If a shape is not a square, then it is not a rectangle. The inverse is false. A 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square, but it is a rectangle. The contrapositive is the negation of the hypothesis and conclusion of the converse. Contrapositive: If a shape is not a rectangle, then it is not a square. The contrapositive is true.
15 Write the converse, inverse, and contrapositive of the statement The sum of the measures of two complementary angles is 90. Determine whether each statement is true or false. If a statement is false, give a counterexample. Answer: Conditional: If two angles are complementary, then the sum of their measures is 90; true. Converse: If the sum of the measures of two angles is 90, then they are complementary; true. Inverse: If two angles are not complementary, then the sum of their measures is not 90; true. Contrapositive: If the sum of the measures of two angles is not 90, then they are not complementary; true.
16 Lesson Quiz: Write the converse, inverse, and contrapostive of the conditional statement If an animal is a cat, then it has four paws. Find the truth value of each. Converse: If an animal has 4 paws, then it is a cat. There are other animals that have 4 paws that are not cats, so the converse is false. Inverse: If an animal is not a cat, then it does not have 4 paws. There are animals that are not cats that have 4 paws, so the inverse is false. Contrapositive: If an animal does not have 4 paws, then it is not a cat; True. Cats have 4 paws, so the contrapositive is true.
17 What did you learn today? How to: Write the converse, inverse, and contrapositive of if-then statements.
18 Assignment: Page , 40 45, 61, 64
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