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. Conditinal: If an integer is divisible by 100, then its last two digits are zeros. Converse: If the last two digits of a number are zeros, then the number is divisible by 100. Biconditional: 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. Conditional: If planes are parallel, then they do not intersect. Converse: If planes do not intersect, then they are parallel. Biconditional: Planes are parallel if and only if they 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. Conditional: If a shape is a rectangle, then it is a four- sided figure with at least 1 right angle. Converse: If a shape is four-sided and has at least 1 right angle, then it is a rectangle. Biconditional:A shape is a rectangle if and only if it 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.