1.5 Related Conditionals

Transcription

1 Name Class Date 1.5 Related Conditionals Essential Question: How are conditional statements related to each other? Explore G.4.B Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse. Also G.4.C Exploring Conditional Statements The conditional statement If p, then q can be written as p q. The notation ~p is used to write not p or the negation of p. You can use Venn diagrams to study conditionals. Resource Locker Shade the region in the diagram that represents the statement. If a figure is a square, then it is a quadrilateral. p q p B q squares squares quadrilaterals quadrilaterals ~p ~q squares squares quadrilaterals quadrilaterals Reflect 1. If you know p q is true and you also know q is true, can you conclude that p is true or that p is false? Explain how the Venn diagram supports your answer. 2. If you know p q is true and you also know p is false, can you conclude that q is true or that q is false? Explain how the Venn diagram supports your answer. 3. If you know p q is true and you know ~q is true, then what can you conclude? Module 1 61 Lesson 5

2 Explain 1 Deciding if the Converse of a Statement is True Converse of a Conditional The converse is the statement formed by exchanging the hypothesis and the conclusion. Conditional: If p, then q. p q Converse: If q, then p. q p Example 1 Write the converse of the statement. Then determine if it is true. If an animal is a dachshund, then it is a dog. The hypothesis is an animal is a dachshund. The conclusion is the animal is a dog. Exchange the hypothesis and conclusion and write the converse. If an animal is a dog, then it is a dachshund. The converse is false. Beagles are dogs, and they are not dachshunds. B If a polygon is a triangle, then it has three sides. Write the hypothesis of the statement. Write the conclusion of the statement. Exchange the hypothesis and conclusion and write the converse. Is the converse true or false? Reflect 4. In the Explore, you used Venn diagrams to study the statement If a figure is a square, then it is a quadrilateral. How does the Venn diagram show that the converse is false? Your Turn 5. Rewrite the statement as a conditional statement. Then write the converse of the conditional. Tell whether the converse is true. We will not go swimming in the lake if it is raining. Image Credits: GK Hart/ Vikki Hart/Photodisc/Getty Images Module 1 62 Lesson 5

3 Explain 2 Deciding if the Inverse or Contrapositive of a Conditional Statement is True Inverse of Conditional The inverse is the statement formed by negating the hypothesis and the conclusion. Conditional: If p, then q. p q Converse: If not p, then not q. ~p ~q Contrapositive of Conditional The contrapositive is the statement formed by exchanging and negating the hypothesis and the conclusion. Conditional: If p, then q. p q Converse: If not q, then not p. ~q ~p Example 2 Write the inverse and the contrapositive of the statement. Then determine if either statement is true. If two angles form a linear pair, then they are supplementary. Write the negation of the hypothesis: Two angles do not form a linear pair. Write the negation of the conclusion: They are not supplementary. Formulate the inverse of the statement: If two angles do not form a linear pair, then they are not supplementary. Formulate the contrapositive of the statement: If two angles are not supplementary, then they do not form a linear pair. The inverse is false. Angles that are not a linear pair can be supplementary. The contrapositive is true. Only supplementary angles can form a linear pair. Module 1 63 Lesson 5

4 B If a triangle is right, then it has exactly two acute angles. Write the negation of the hypothesis: Write the negation of the conclusion: Formulate the inverse of the statement: Formulate the contrapositive of the statement: The inverse is The contrapositive is. All obtuse triangles have exactly one/ two/ three acute angles.. If all three angles are acute, the triangle can/cannot be right. Reflect 6. If two angles form a linear pair, then they are supplementary. Your friend writes the negation of the conclusion as the two angles are complementary. Is this correct? Explain. Your Turn Use the conditional statement: If it is Friday, then tomorrow is Saturday. 7. Write the inverse of the statement. 8. Write the contrapositive of the statement. Explain 3 Biconditional Writing Biconditional Statements The biconditional is a statement that can be written in the form p if and only if q. The statement is equivalent to p q and q p, and can be written as p q. Module 1 64 Lesson 5

5 Example 3 Write the converse and the biconditional of the statement. Is the converse or the biconditional true? If M is the midpoint of _ AC, then _ AM and _ CM have the same length. Exchange the hypothesis and conclusion and write the converse: _ If AM and BM _ have the same length, then M is the midpoint of AC _. Write the biconditional: _ AM and _ CM have the same length, if and only if M is the midpoint of _ AC. The converse is false. _ AM and _ CM could have the same length, without A, C, and M being collinear. The biconditional is false. It represents both the original statement and the converse, so if either of those statements is false, the biconditional is false. B If a + b = 0, then a and b are opposites. Exchange the hypothesis and conclusion and write the converse: Write the biconditional: The converse is. If a and b are opposites, then a = - b and a + b = 0. The biconditional is. It represents the original statement and the converse, both of which are. Reflect 9. Discussion How can the word biconditional help you remember what a biconditional is? Your Turn 10. Consider the biconditional statement: a transformation is a rigid motion if and only if lengths and angle measures are preserved. Explain how this statement can be written as a conditional and its converse. Module 1 65 Lesson 5

6 Elaborate 11. If a conditional statement is true, is its contrapositive also true? Explain using a Venn diagram. 12. If a conditional statement is false, is its contrapositive also false? Explain using a Venn diagram. 13. Essential Question Check-In If you know that a conditional statement is true, do you know whether its converse is true? Do you know whether its contrapositive is true? Evaluate: Homework and Practice Write the converse and determine if it is true. If not, give a counterexample. 1. If x > y, then x > y. 2. If the season is winter, then the month is January. 3. If an object is a pencil, then it is used to write. Online Homework Hints and Help Extra Practice Image Credits: Jose Luis Pelaez Inc./Blend Images/age fotostock 4. If it s later than 5 p.m., it s later than 4 p.m. Module 1 66 Lesson 5

7 Write the inverse and determine if it is true. If not, give a counterexample. 5. If two lines are parallel, then they do not intersect. 6. If a number is an integer, then it is a whole number. 7. If an angle is obtuse, then it is not acute. 8. If x = 2, then x 2 = 4. Write the contrapositive and determine if it is true. If not, give a counterexample. 9. If a letter is a vowel, then it precedes t in the alphabet. 10. If the converse of a statement is true, then the inverse of the statement is true. 11. If the radius of a circle is 10, then the diameter is If four points are collinear, then they are coplanar. Write a biconditional for the statement. 13. A segment bisector is a ray, segment, or line that divides a segment into two congruent segments. 14. A circle is the set of all points in a plane that are a fixed distance from a given point. 15. A vehicle must have a parking decal to park in the visitor s lot. Module 1 67 Lesson 5

8 Identify a conditional and its converse in the biconditional. 16. Tom will bring an umbrella if and only if the forecast calls for rain. 17. A rectangle is a square if and only if it has four congruent sides. 18. Two planes do not intersect if and only if they are parallel. 19. Multi-Step The following statement is true: If point A is at (2, 4) and point B is at (6, 10), then the midpoint M of _ AB is (4, 7). Show that the converse is false. Explain your solution. 20. Communicate Mathematical Ideas A friend says that given a conditional statement, if you take the converse, and then take the inverse of the converse, it is equivalent to taking the contrapositive of the original statement. Do you agree? Explain your answer using the notation q p to represent the original conditional. 21. Given the statement: If a blob is slimy, then it is gooey. If this statement is true, which of the following statements must be true? Select all that apply. A. If a blob is not slimy, it is not gooey. B. If an object is gooey, then it is a blob. C. If a blob is not gooey, then it is not slimy. D. If a blob is gooey, then it is slimy. E. A blob is gooey, if and only if it is slimy. Module 1 68 Lesson 5

9 H.O.T. Focus on Higher Order Thinking 22. Analyze Relationships Consider the statement: if a number is divisible by m, then it is divisible by n. Describe the relationship between m and n if the statement is true. Then consider the same question applied to the inverse of the statement. Assume that m and n are integers greater than Communicate Mathematical Ideas If a transformation is a rigid motion, then it is a translation. Determine if the original statement, the converse, the inverse, and the contrapositive are true. Explain your reasoning. 24. Justify Reasoning If the city places a new stop light at the intersection of West Street and Arbor Road, then traffic heading into the square will increase. You notice traffic heading into the square has increased. Can you assume that the city placed the new stop light? Explain in terms of related conditionals. Image Credits: (t) ImageGap/iStockPhoto.com; (b) Hero Images/Corbis 25. Multiple Representations At a health club, members can use weights or cardio equipment, among other amenities. Let p = a member uses weights and q = a member uses cardio equipment. Write a biconditional statement that describes the intersection of the circles in the Venn diagram. p q Module 1 69 Lesson 5

10 Lesson Performance Task John Venn designed this Venn diagram in the 1880s. Here, each oval represents numbers that are multiples of the indicated number. If you draw horizontal lines through a section that represents a hypothesis p, and then vertical lines through the section that represents a conclusion q, you can determine whether p q and related conditionals are true. Use the technique given, and what you know about multiples, to determine whether the following statements are true: multiple of 2 multiple of 3 multiple of 5 multiple of 7 If a number is a multiple of 6, then it is a multiple of 3. If a number is a multiple of 21, then it is a multiple of 42. If a number is a not multiple of 7, then it is a not multiple of 21. Write a conjecture about determining whether p q is true, strictly in terms of using the line drawing technique and the diagram. Module 1 70 Lesson 5