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.

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 x3 = –27.

2-2

Biconditionals and Biconditionals and DefinitionsDefinitions

Biconditionals and Biconditionals and DefinitionsDefinitions

Section 2-2Section 2-2

Objectives• To write biconditionals.

• To recognize good definitions.

ObjectiveA ______________ is the combination of a conditional statement and its converse.

A biconditional (statement) contains the words “___________________.”

This is an if-then statement called a _______________.

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.

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.

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.

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.

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.

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.

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.

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.

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.

A good definition is a statement thatcan help you to ____________ or ___________ an object.

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.

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.

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).

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.

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.

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.

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.