1. Use the following conditional for Exercises 1–3. If a circle’s radius is 2 m, then its diameter is 4 m. 1.Identify the hypothesis and conclusion. Hypothesis: A circle’s radius is 2 m. Conclusion: Its diameter is 4 m. 2.Write the converse. If a circle’s diameter is 4 m, then its radius is 2 m. 3.Determine the truth value of the conditional and its converse. Both are true. Show that each conditional is false by finding a counterexample. 4.If lines do not intersect, then they are parallel. skew lines 5.All numbers containing the digit 0 are divisible by 10. Sample: 105 Conditional Statements Hypothesis: A circle’s radius is 2 m. Conclusion: Its diameter is 4 m. If a circle’s diameter is 4 m, then its radius is 2 m. Both are true. skew lines Sample: 105 GEOMETRY LESSON 2-1 2-1

2. Biconditionals and Definitions Identify the hypothesis and the conclusion of each conditional statement. 1.If x > 10, then x > 5. 2.If you live in Milwaukee, then you live in Wisconsin. Write each statement as a conditional. 3.Squares have four sides.4.All butterflies have wings. Write the converse of each statement. 5.If the sun shines, then we go on a picnic. 6.If two lines are skew, then they do not intersect. 7.If x = –3, then x 3 = –27. (For help, go to the Lesson 2-1.) GEOMETRY LESSON 2-2 2-2 Check Skills You’ll Need

3. Biconditionals and Definitions 1.The hypothesis follows if and the conclusion follows then; so the hypothesis is “x > 10” and the conclusion is “x > 5.” 2.The hypothesis follows if and the conclusion follows then; so the hypothesis is “you live in Milwaukee” and the conclusion is “you live in Wisconsin.” 3.Rewrite the statement as an if-then statement: If a figure is a square, then it has four sides. 4.Rewrite the statement as an if-then statement: If something is a butterfly, then it has wings. 5.Switch the hypothesis and conclusion: If we go on a picnic, then the sun shines. 6.Switch the hypothesis and conclusion: If two lines do not intersect, then they are skew. 7.Switch the hypothesis and conclusion: If x 3 = –27, then x = –3. GEOMETRY LESSON 2-2 Solutions 2-2

4. Biconditionals and Definitions GEOMETRY LESSON 2-2 2-2 When you combine a conditional statement and its converse, you create a biconditional statement. A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.”

5. Biconditionals and Definitions GEOMETRY LESSON 2-2 2-2 p q means p q and q p The biconditional “p if and only if q” can also be written as “p iff q” or p  q. Writing Math

6. Biconditionals and Definitions GEOMETRY LESSON 2-2 2-2 For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false.

7. Biconditionals and Definitions GEOMETRY LESSON 2-2 2-2

8. Biconditionals and Definitions GEOMETRY LESSON 2-2 2-2 A good definition is a statement that can help you identify or classify an object. A good definition has several important components. A good definition uses clearly understood terms. The terms should be commonly understood or already defined. A good definition is precise. Good definitions avoid words such as large, sort of, and almost. A good definition is reversible. That means that you can write a good definition as a true biconditional.

9. Biconditionals and Definitions Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional: If x = 5, then x + 15 = 20. To write the converse, exchange the hypothesis and conclusion. Converse: If x + 15 = 20, then x = 5. When you subtract 15 from each side to solve the equation, you get x = 5. Because both the conditional and its converse are true, you can combine them in a true biconditional using the phrase if and only if. Biconditional: x = 5 if and only if x + 15 = 20. GEOMETRY LESSON 2-2 2-2 Quick Check

10. Biconditionals and Definitions Write the two statements that form this biconditional. A biconditional is written as two conditionals that are converses of each other. Conditional: If lines are skew, then they are noncoplanar. Converse: If lines are noncoplanar, then they are skew. Biconditional: Lines are skew if and only if they are noncoplanar. GEOMETRY LESSON 2-2 2-2 Quick Check

11. Biconditionals and Definitions Show that this definition of triangle is reversible. Then write it as a true biconditional. The original conditional is true. Conditional: If a polygon is a triangle, then it has exactly three sides. The converse is also true. Converse: If a polygon has exactly three sides, then it is a triangle. Because both statements are true, they can be combined to form a biconditional. A polygon is a triangle if and only if it has exactly three sides. Definition: A triangle is a polygon with exactly three sides. GEOMETRY LESSON 2-2 2-2 Quick Check

12. Biconditionals and Definitions Is the following statement a good definition? Explain. The statement is true as a description of an apple. Now exchange “An apple” and “a fruit that contains seeds,” and the reverse reads: A fruit that contains seeds is an apple. There are many other fruits containing seeds that are not apples, such as lemons and peaches. These are counterexamples, so the reverse of the statement is false. The original statement is not a good definition because the statement is not reversible. An apple is a fruit that contains seeds. GEOMETRY LESSON 2-2 2-2 Quick Check

13. 1.Write the converse of the statement. If it rains, then the car gets wet. 2.Write the statement above and its converse as a biconditional. 3.Write the two conditional statements that make up the biconditional. Lines are skew if and only if they are noncoplanar. Is each statement a good definition? If not, find a counterexample. 4.The midpoint of a line segment is the point that divides the segment into two congruent segments. 5.A line segment is a part of a line. Biconditionals and Definitions If the car gets wet, then it rains. It rains if and only if the car gets wet. If lines are skew, then they are noncoplanar; if lines are noncoplanar, then they are skew. yes No; the statement is not reversible; a ray. GEOMETRY LESSON 2-2 2-2