Statements Goals Identify and evaluate conditional statements. Identify converses and biconditionals. Drafting, Sports, Geography

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1 3-6 Conditional Statements Goals Identify and evaluate conditional statements. Identify converses and biconditionals. Applications Drafting, Sports, Geography Do you think each statement is true or false? Explain your reasoning. a. Denver is the capital of Colorado. b. For all x, x 2 x. c. Lines m and n are parallel. BUILD UNDERSTANDING Many of the statements in this chapter are written in if then form. Statements like these are called conditional statements, or simply conditionals. The clause following if is called the hypothesis of the conditional. The clause following then is called the conclusion. For example, the parallel lines postulate was presented as a conditional. Denver, C If two parallel lines are cut by a transversal, hypothesis then corresponding angles are equal in measure. conclusion A conditional is either true or false. When a conditional is true, you can justify it in a variety of ways. For instance, you may be able to show that the conditional is true because it follows directly from a definition. When a conditional is a postulate, such as the parallel lines postulate, it is assumed to be true. Still other conditionals are theorems, and these must be proved true. To demonstrate that a conditional is false, you need to find only one example for which the hypothesis is true but the conclusion is false. An example like this is called a counterexample. E x a m p l e 1 Tell whether each conditional is true or false. a. If two lines are parallel, then they are coplanar. b. If two lines do not intersect, then they are parallel. a. arallel lines are defined as coplanar lines that do not intersect. So, the conditional is true. b. Consider skew lines k and shown. By the definition of skew lines, k and do not intersect, and so the hypothesis is true. However, also by the definition of skew lines, k and are noncoplanar. Lines k and cannot be parallel, and so the conclusion is false. Therefore, lines k and are a counterexample, and the conditional is false. k 128 Chapter 3 Geometry and Reasoning

2 The converse of a conditional is formed by interchanging the hypothesis and the conclusion. The fact that a conditional is true is no guarantee that its converse is true. Reading Math E x a m p l e 2 DRAFTING eople who draw plans must apply this true statement: If two lines are parallel, then they do not intersect. Write the converse of the statement. Is it also true? Interchange the hypothesis and the conclusion of the given statement. Statement: If two lines are parallel, then they do not intersect. hypothesis conclusion Converse: If two lines do not intersect, then they are parallel. Many statements about everyday situations can be expressed as conditionals. For instance, the following is a true conditional. If it is raining, then it is cloudy. Its converse is false. If it is cloudy, then it is raining. hypothesis conclusion By definition, parallel lines do not intersect, and so the given statement is true. art b of Example 1 demonstrated that lines that do not intersect are not necessarily parallel, and so the converse is false. The converse of the parallel lines postulate also is assumed to be true. It is stated as the corresponding angles postulate in the following manner. ostulate 10 The Corresponding Angles ostulate If two lines are cut by a transversal so that a pair of corresponding angles are equal in measure, then the lines are parallel. When a statement and its converse are both true, they can be combined into an if and only if statement. This type of statement is called a biconditional statement, or simply a biconditional. Every definition can be written as a biconditional. E x a m p l e 3 Write this definition as two conditionals and as a single biconditional. A right angle is an angle whose measure is 90. The definition leads to two true conditionals. If an angle is a right angle, then its measure is 90. If the measure of an angle is 90, then it is a right angle. These can be combined into a single biconditional as follows. An angle is a right angle if and only if its measure is 90. roblem Solving Tip When writing the converse of a conditional, you may need to change the wording of the hypotheses and the conclusion to make the converse read clearly. mathmatters3.com/extra_examples Lesson 3-6 Conditional Statements 129

3 TRY THESE EXERCISES 1. TALK ABUT IT Decide whether this conditional is true or false. If two lines are each perpendicular to a third line, then they are parallel to each other. Discuss your reasoning with a classmate. 2. Write the converse of this statement. If two angles are vertical angles, then they are equal in measure. Are the given statement and its converse true or false? 3. Write this definition as two conditionals and as a single biconditional. The bisector of an angle is the ray that divides the angle into two adjacent angles that are equal in measure. 4. NUMBER SENSE Tell whether this conditional is true or false. If a number is less than 1, the number is a proper fraction. Write the converse of the statement. Is the converse true or false? 5. SRTS If a shortstop makes a bad throw to first base, the error is charged to the shortstop. This statement is true. Write the converse of the statement. Is it true or false? 6. GEGRAHY If a point is located north of the equator, it has a northern latitude. This statement is true. Write the converse of the statement. Is it true or false? RACTICE EXERCISES For Extra ractice, see page 672. Sketch a counterexample that shows why each conditional is false. 7. If line t intersects lines g and h, then line t is a transversal. 8. If Q QR, then point Q is the midpoint of R. 9. If points A, B, and C are collinear, then BA and BC are opposite rays. 10. If two angles share a common side and a common vertex, then they are adjacent angles. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 11. If points J, K, and L are coplanar, then they are collinear. 12. If point Y is the midpoint of X Z, then XY YZ XZ. 13. If the sum of the measures of two angles is 90, then the angles are complementary. 14. If two angles are supplementary, then the sum of their measures is greater than If two lines are perpendicular, then they do not intersect. 16. If m QRS m SRT, then RS bisects QRT. 130 Chapter 3 Geometry and Reasoning

4 Write each definition as two conditionals and as a single biconditional. 17. The midpoint of a segment is the point that divides it into two segments of equal length. 18. erpendicular lines are two lines that intersect to form right angles. 19. A transversal is a line that intersects two or more coplanar lines in different points. 20. Vertical angles are two angles whose sides form two pairs of opposite rays. GEMETRIC CNSTRUCTIN The corresponding angles postulate provides a method for constructing parallel lines. In the figure at the right, you see the finished construction of a line parallel to line through point. Trace line and point onto a sheet of paper and repeat the construction. Then complete the statements below that outline the steps of the construction. 21. Step 1: Using a straightedge, draw any line? through point intersecting line. Label the intersection point?. 22. Step 2: With the compass point at?, draw an arc intersecting lines t and. Label the intersection points? and?. 23. Step 3: Using the same radius as in Step 2, place the compass point at point? and draw an arc intersecting line t. Label the intersection point?. 24. Step 4: lace the compass point at point? and the pencil at point?. Using this radius, draw an arc that intersects line. 25. Step 5: Using the same radius as in Step 4, place the compass point at point? and draw an arc that intersects the arc you drew in Step 3. Label the intersection point?. 26. Step 6: Draw line? through points and Y. m? m?, and so??. m t X Y A Q B EXTENDED RACTICE EXERCISES WRITING MATH Explain why each of the following is not a good definition. 27. Vertical angles are two angles whose sides form opposite rays. 28. A line segment is part of a line. 29. Complementary angles are adjacent angles whose exterior sides form a right angle. 30. Skew lines are noncoplanar lines that do not intersect. MIXED REVIEW EXERCISES Find each length. (Lesson 3-1) 31. In the figure below, AC 130. Find BC. 32. In the figure below, LM 94. Find LN. 3x 4 4x 7p 14 3p L M A B C N 33. In the figure below, JL 88. Find KL. 34. In the figure below, QS 41. Find QR. 3y 8 9y 2z 14 3z 18 J K L Q R S mathmatters3.com/self_check_quiz Lesson 3-6 Conditional Statements 131

5 Name Date RETEACHING 3-6 CNDITINAL STATEMENTS Conditional statements are written in an if-then (hypothesis/conclusion) form. The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion. A counterexample proves a conditional or converse is false. E x a m p l e Write the converse of this statement: If RS TU, then RS, TU, and RT are coplanar. Then decide whether the statement and its converse are true or false. If false, give a counterexample. Converse: If R S, T U and R T are coplanar, then R S T U. riginal statement is true, since parallel lines are coplanar by definition and for any two points in a plane, the line joining them lies in the plane (ostulate 3). Converse is false. Counterexample: Each pair of coplanar lines could intersect to form a triangle. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If false, give a counterexample. 1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D are collinear. 2. If m IQJ m HQJ 180, then IQJ and HQJ are obtuse angles. 3. If three lines have one point in common, then they are coplanar. 4. If two lines are skew, then they are not coplanar. Glencoe/McGraw-Hill 88 MathMatters 3

6 Name Date EXTRA RACTICE 3-6 CNDITINAL STATEMENTS Sketch a counterexample that shows why each conditional is false. Use your own paper. 1. If point A is the midpoint of C D, then C A D A. 2. If XY and XZ are opposite rays, then point X is the midpoint of Y Z. 3. If RST and TSV are congruent, then ST bisects RSV. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 4. If points A, B, and C are collinear, then B is the midpoint of A C. 5. If point X is the vertex of 1 and 2, then 1 and 2 are adjacent angles. 6. If two angles are complementary, then both of the angles are acute. 7. If two lines intersect, then they are parallel. 8. If an angle is obtuse, then its supplement is acute. Write each definition as two conditionals and as a single biconditional. 9. Coplanar points are points that lie in the same plane. 10. A segment is a part of a line that begins at one endpoint and ends at another. Glencoe/McGraw-Hill 89 MathMatters 3

7 Review and ractice Your Skills RACTICE LESSN 3-5 Draw the next figure in each pattern. Then describe the tenth figure in the pattern RACTICE LESSN 3-6 Sketch a counterexample to show why each conditional is false. 9. If ABC and DEF are supplements, then m ABC m DEF. 10. If three points are coplanar, then they are collinear. 11. If two lines are skew, then they intersect. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 12. If two lines intersect, then they are perpendicular. 13. If C is the midpoint of A B, then AB 2(AC ). 14. If two angles are vertical angles, then their supplements are equal. Write each definition as two conditionals and as a single biconditional. 15. erpendicular lines are lines that intersect to form right angles. 16. Skew lines are noncoplanar lines. 132 Chapter 3 Geometry and Reasoning

8 Name Date ENRICHMENT 3-6 CATEGRICAL RSITINS A categorical proposition is a statement about an entire category or class of things. There are four different standard forms of categorical propositions. All S is. No S is. Some S is. Some S is not. All dogs are friendly. No dogs are friendly. Some dogs are friendly. Some dogs are not friendly. Venn diagrams can be used to illustrate categorical propositions. E x a m p l e 1 Diagram All S is. E x a m p l e 2 Diagram Some S is not. The shading shows that this part of the diagram has no members. The X shows that this part of the diagram has at least one member. Draw a Venn diagram for each categorical proposition. 1. No S is. 2. Some S is. Write the converse of each of the four standard forms. Then draw a Venn diagram for each one Which of the standard forms are logically equivalent to their converses? Glencoe/McGraw-Hill 90 MathMatters 3

10 The converse of a conditional is formed by interchanging the hypothesis and the conclusion. The fact that a conditional is true is no guarantee that its converse is true. E x a m p l e 2 DRAFTING eople who draw plans must apply this true statement: If two lines are parallel, then they do not intersect. Write the converse of the statement. Is it also true? Interchange the hypothesis and the conclusion of the given statement. Statement: If two lines are parallel, then they do not intersect. hypothesis conclusion Converse: If two lines do not intersect, then they are parallel. hypothesis conclusion By definition, parallel lines do not intersect, and so the given statement is true. art b of Example 1 demonstrated that lines that do not intersect are not necessarily parallel, and so the converse is false. The converse of the parallel lines postulate also is assumed to be true. It is stated as the corresponding angles postulate in the following manner. ostulate 10 The Corresponding Angles ostulate If two lines are cut by a transversal so that a pair of corresponding angles are equal in measure, then the lines are parallel. Reading Math Many statements about everyday situations can be expressed as conditionals. For instance, the following is a true conditional. If it is raining, then it is cloudy. Its converse is false. If it is cloudy, then it is raining. Chalkboard Examples Supplementary Example 1 Tell whether this conditional is true or false: If two lines are coplanar, then they are not skew lines. true Supplementary Example 2 Write the converse of this statement: If a number is divisible by six, then it is divisible by three. Then decide whether the statement and its converse are true or false. If a number is divisible by three, then it is divisible by six. The statement is true; the converse is false. Supplementary Example 3 Write this definition as two conditionals and a single biconditional: A triangle is a polygon with three sides. If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is a triangle. A polygon is a triangle if and only if it has three sides. When a statement and its converse are both true, they can be combined into an if and only if statement. This type of statement is called a biconditional statement, or simply a biconditional. Every definition can be written as a biconditional. E x a m p l e 3 Write this definition as two conditionals and as a single biconditional. A right angle is an angle whose measure is 90. The definition leads to two true conditionals. If an angle is a right angle, then its measure is 90. If the measure of an angle is 90, then it is a right angle. These can be combined into a single biconditional as follows. An angle is a right angle if and only if its measure is 90. mathmatters3.com/extra_examples roblem Solving Tip When writing the converse of a conditional, you may need to change the wording of the hypotheses and the conclusion to make the converse read clearly. Lesson 3-6 Conditional Statements 129 Reteaching Worksheet 3-6 Name RETEACHING 3-6 Date CNDITINAL STATEMENTS Conditional statements are written in an if-then (hypothesis/conclusion) form. The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion. A counterexample proves a conditional or converse is false. E x a m p l e Write the converse of this statement: If RS TU, then RS, TU, and RT are coplanar. Then decide whether the statement and its converse are true or false. If false, give a counterexample. Converse: If R S, T U and R T are coplanar, then R S T U. riginal statement is true, since parallel lines are coplanar by definition and for any two points in a plane, the line joining them lies in the plane (ostulate 3). Converse is false. Counterexample: Each pair of coplanar lines could intersect to form a triangle. 11. Converse; If points J, K, and L are collinear, then they are coplanar. The given statement is false. Its converse is true. 12. Converse: If XY YZ XZ, then point Y is the midpoint of XZ. The given statement is true. Its converse is false. 13. Converse: If two angles are complementary, then the sum of their measures is 90º. Both the given statement and its converse are true. 14. Converse: If the sum of the measures of two angles is greater than 90º, then the angles are supplementary. The given statement is true. Its converse is false. 15. Converse: If two lines do not intersect, then they are perpendicular. Both the given statement and its converse are false. 16. Converse: If RS bisects QRT, then m QRS m SRT. The given statement is false. Its converse is true. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If false, give a counterexample. 1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D are collinear. If points A, B, C, and D are collinear, then points A, B, C, and D lie in both plane L and plane M.; true; false; Counterexample: AD lies on plane L and plane L is parallel to plane M. 2. If m IQJ m HQJ 180, then IQJ and HQJ are obtuse angles. If IQJ and HQJ are obtuse angles, then m IQJ m HQJ 180.; false; Counterexample: If m IQJ 30 and m HQJ 170, then m IQJ m HQJ ; true 3. If three lines have one point in common, then they are coplanar. If three lines are coplanar, then they have one point in common. false; Counterexample: 2 lines can be coplanar and a third line can intersect that plane at only one point.; false; 3 parallel lines 4. If two lines are skew, then they are not coplanar. If two lines are not coplanar, then they are skew.; true; true Lesson 3-6 Conditional Statements 129

13 Skills ractice Vocabulary Review Lesson 3-5 inductive reasoning conjecture sequence Lesson 3-6 conditional conclusion biconditional ASSIGNMENT GUIDE All students: 1 24 hypothesis counterexample ADDITINAL ANSWERS Review and ractice Your Skills RACTICE LESSN 3-5 Draw the next figure in each pattern. Then describe the tenth figure in the pattern. See additional answers for drawings units horizontal; 10 units vertical units long same orientation as second figure, but 512 individual angles, each with ten lines in the interior same as second figure 512 units each as long as original line. same as second figure 5. RACTICE LESSN sided polygon with all diagonals C D A B E F Y X Z Sketch a counterexample to show why each conditional is false. See additional answers for sample answers. 9. If ABC and DEF are supplements, then m ABC m DEF. 10. If three points are coplanar, then they are collinear. 11. If two lines are skew, then they intersect. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If two lines are perpendicular, then they 12. If two lines intersect, then they are perpendicular. intersect. false, true 13. If C is the midpoint of A B, then AB 2(AC ). If AB 2(AC), then C is the midpoint of A B. true, false 14. If two angles are vertical angles, then their supplements are equal. If the supplements of two angles are equal, then they are vertical angles. true, false Write each definition as two conditionals and as a single biconditional. 15. erpendicular lines are lines that intersect to form right angles. See additional answers. 16. Skew lines are noncoplanar lines. If lines are skew, they are noncoplanar. If lines are noncoplanar, then they are skew. Two lines are skew if and only if they are noncoplanar. Chapter 3 Geometry and Reasoning 15. If two lines are perpendicular, then they intersect to form right angles. If two lines intersect to form right angles, then they are perpendicular. Two lines are perpendicular if and only if they intersect to form right angles. 132 Chapter 3 Geometry and Reasoning

14 Answers (Lesson 3-5 and 3-6) Name Date ENRICHMENT 3-5 ATTERNS WITH DISSECTINS There are five ways to dissect a convex five-sided polygon into three triangles. 1. Show all the different ways to dissect a convex four-sided polygon into 2 triangles. How many are there? 2 ways 2. Show all the different ways to dissect a convex six-sided polygon into 4 triangles. How many are there? 14 ways Study the patterns in this chart. Then fill in the missing values Number Number of of Sides Dissections Formula for D n 3 D D 4 2 D 3 D 3 5 D 5 5 D 4 D 3 D 3 D 4 6 D 6 14 D 5 D 3 D 4 D 4 D 3 D 5 7 D 7 42 D 6 D 3 D 5 D 4 D 4 D 5 D 3 D 6 8 D D 7 D 3 D 6 D 4 D 5 D 5 D 4 D 6 D 3 D 7 9 D D 8 D 3 D 7 D 4 D 6 D 5 D 5 D 6 D 4 D 7 D 3 D 8 10 D D 9 D 3 D 8 D 4 D 7 D 5 D 6 D 6 D 5 D 7 D 4 D 8 D 3 D 9 Glencoe/McGraw-Hill 87 MathMatters 3 Name Date RETEACHING 3-6 CNDITINAL STATEMENTS Conditional statements are written in an if-then (hypothesis/conclusion) form. The converse of a conditional statement is formed by interchanging the hypothesis and the conclusion. A counterexample proves a conditional or converse is false. E x a m p l e Write the converse of this statement: If RS TU, then RS, TU, and RT are coplanar. Then decide whether the statement and its converse are true or false. If false, give a counterexample. Converse: If R S, T U and R T are coplanar, then R S T U. riginal statement is true, since parallel lines are coplanar by definition and for any two points in a plane, the line joining them lies in the plane (ostulate 3). Converse is false. Counterexample: Each pair of coplanar lines could intersect to form a triangle. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. If false, give a counterexample. 1. If points A, B, C and D lie on both plane L and plane M, then points A, B, C, and D are collinear. If points A, B, C, and D are collinear, then points A, B, C, and D lie in both plane L and plane M.; true; false; Counterexample: AD lies on plane L and plane L is parallel to plane M. 2. If m IQJ m HQJ 180, then IQJ and HQJ are obtuse angles. If IQJ and HQJ are obtuse angles, then m IQJ m HQJ 180.; false; Counterexample: If m IQJ 30 and m HQJ 170, then m IQJ m HQJ ; true 3. If three lines have one point in common, then they are coplanar. If three lines are coplanar, then they have one point in common. false; Counterexample: 2 lines can be coplanar and a third line can intersect that plane at only one point.; false; 3 parallel lines 4. If two lines are skew, then they are not coplanar. If two lines are not coplanar, then they are skew.; true; true Glencoe/McGraw-Hill 88 MathMatters 3 Glencoe/McGraw-Hill A9 MathMatters 3

15 Answers (Lesson 3-6) Name Date EXTRA RACTICE 3-6 CNDITINAL STATEMENTS Sketch a counterexample that shows why each conditional is false. Use your own paper. Check students drawings. 1. If point A is the midpoint of C D, then C A D A. 2. If XY and XZ are opposite rays, then point X is the midpoint of Y Z. 3. If RST and TSV are congruent, then ST bisects RSV. Write the converse of each statement. Then tell whether the given statement and its converse are true or false. 4. If points A, B, and C are collinear, then B is the midpoint of A C. midpoint of A C, then points A, B, and C are collinear; false; true 5. If point X is the vertex of 1 and 2, then 1 and 2 are adjacent angles. and 2 are adjacent angles, then they have a common vertex; false; true 6. If two angles are complementary, then both of the angles are acute. angles are acute, then the angles are complementary.; true; false 7. If two lines intersect, then they are parallel. the line intersect.; false; false 8. If an angle is obtuse, then its supplement is acute. angle is acute, then the angle is obtuse.; true; true If B is the If two If 1 If two lines are parallel, then If the supplement of an Write each definition as two conditionals and as a single biconditional. 9. Coplanar points are points that lie in the same plane. If points are coplanar, then they lie in the same plane. If points lie in the same plane, then the points are coplanar. oints are coplanar if and only if they lie in the same plane 10. A segment is a part of a line that begins at one endpoint and ends at another. If a figure is a segment, then it is a part of a line that begins at one endpoint and ends at another. If a figure is a part of a line that begins at one endpoint and ends at another, then the figure is a segment. A figure is a segment if and only if it is part of a line that begins at one endpoint and ends at another. Glencoe/McGraw-Hill 89 MathMatters 3 Name Date ENRICHMENT 3-6 CATEGRICAL RSITINS A categorical proposition is a statement about an entire category or class of things. There are four different standard forms of categorical propositions. All S is. All dogs are friendly. No S is. No dogs are friendly. Some S is. Some dogs are friendly. Some S is not. Some dogs are not friendly. Venn diagrams can be used to illustrate categorical propositions. E x a m p l e 1 E x a m p l e 2 Diagram All S is. Diagram Some S is not. The shading shows that this part of the diagram has no members. The X shows that this part of the diagram has at least one member. Draw a Venn diagram for each categorical proposition. 1. No S is. 2. Some S is. S S X Write the converse of each of the four standard forms. Then draw a Venn diagram for each one. All is S S No is S. S Some is S Some is not S. S X 7. Which of the standard forms are logically equivalent to their converses? No S is. No is S. and Some S is. Some is S. Glencoe/McGraw-Hill 90 MathMatters 3 Glencoe/McGraw-Hill A10 MathMatters 3

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