Section 2-3: Biconditionals & Definitions. Objectives: Write Biconditionals and recognize good definitions. Recall. Form of a Conditional Statement. Biconditional. p q. Write a bi-conditional only if BOTH the conditional and the converse are TRUE .
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Section 2-3: Biconditionals& Definitions
If two angles have the same measure, then the angles are congruent.
If two angles are congruent, then they have the same measure.
Both statements are true, so….
…you can write a Biconditional statement:
Two angles have the same measure if and only if the angles are congruent.
Write a Bi-conditional Statement
Consider the following true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional.
If x = 5, then x + 15 = 20.
If x + 15 = 20, then x = 5.
Since both the conditional and its converse are true, you can combine them in a true biconditional using the phrase if and only if.
x = 5 if and only if x + 15 = 20.
Consider the biconditional statement:
A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
If a number is divisible by 3, then the sum of its digits is divisible by 3.
If the sum of a numbers digits is divisible by 3, then the number is divisible by 3.
Separate a Biconditional
Write the two statements that form this biconditional.
Lines are skew if and only if they are noncoplanar.
If lines are skew, then they are noncoplanar.
If lines are noncoplanar, then they are skew.
Show definition of perpendicular lines is reversible
Perpendicular lines are two lines that intersect to form right angles
If two lines are perpendicular, then they intersect to form right angles.
If two lines intersect to form right angles, then they are perpendicular.
Since both are true converses of each other, the definition can be written as a true biconditional:
“Two lines are perpendicular iff they intersect to form right angles.”
Show that the definition of triangle is reversible. Then write it as a true biconditional.
1. Write the conditional
2. Write the converse
3. Determine if both statements are true
4. If true, combine to form a biconditional.
Definition: A triangle is a polygon with exactly three sides.
If a polygon is a triangle, then it has exactly three sides.
If a polygon has exactly three sides, then it is a triangle.
A polygon is a triangle if and only if it has exactly three sides.
Is the following statement a good definition? Explain.
An apple is a fruit that contains seeds.
Conditional: If a fruit is an apple then it contains seeds.
Converse: If a fruit contains seed then it 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.