1. 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. 2-2

2. Biconditionals and Definitions Section 2-2

3. Objectives To write biconditionals. To recognize good definitions.

4. Objective A ______________ is the combination of a conditional statement and its converse. A biconditional (statement) contains the words “___________________.” This is an if-then statement called a _______________.

5. Consider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. 1. Conditional: If two angles have the same measure, then the angles are congruent.

6. Consider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. 2. Conditional: If three points are collinear, then they lie on the same line.

7. Consider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. 3. Conditional: If two segments have the same length, then they are congruent.

8. Consider the true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. 4. Conditional: If x = 12, then 2x – 5 = 19.

9. Separating a Biconditional into Parts Write the two (conditional) statements that form the biconditional. 1. A number is divisible by three if and only if the sum of its digits is divisible by three.

10. Separating a Biconditional into Parts Write the two (conditional) statements that form the biconditional. 2. A number is prime if and only if it has two distinct factors, 1, and itself.

11. Separating a Biconditional into Parts Write the two (conditional) statements that form the biconditional. 3. A line bisects a segment if and only if the line intersects the segment only at its midpoint.

12. Separating a Biconditional into Parts Write the two (conditional) statements that form the biconditional. 4. An integer is divisible by 100 if and only if its last two digits are zeros.

13. Recognizing a Good Definition Use the examples to identify the figures above that are polyglobs. Write a definition of a polyglob by describing what a polyglob is.

14. A good definition is a statement that can help you to ____________ or ___________ an object.

15. Key components of a good definition 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 some. A good definition is reversible. That means that you can write a good definition as a true biconditional.

16. Show that the definition is reversible. Then write it as a true biconditional. 1. Definition: Perpendicular lines are two lines that intersect to form right angles.

17. Show that the definition is reversible. Then write it as a true biconditional. 2. Definition: A right angle is an angle whose measure is 90 (degrees).

18. Show that the definition is reversible. Then write it as a true biconditional. 3. Definition: Parallel planes are planes that do not intersect.

19. Show that the definition is reversible. Then write it as a true biconditional. 4. Definition: A rectangle is a four-sided figure with at least one right angle.

20. Is the given statement a good definition? Explain. 1.An airplane is a vehicle that flies. 2.A triangle has sharp corners. 3.A square is a figure with four right angles.

21. 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.