identify the hypothesis and the conclusion of each conditional statement. 1.if x > 10, then x...
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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.