Section 2-3: Biconditionals & Definitions
Objectives:
•Write Biconditionals and recognize good definitions
Conditional Statements and Converses
Statement
Example
Symbolic
You read as
Conditional
If an angle is a straight angle,
then its measure is 180º.
p
q
If p, then q.
Converse
If the measure of an angle is
180º, then it is a straight angle.
q
p
If q then p.
Form of a Conditional Statement
p q
• Write a bi-conditional only if BOTH the conditional and the converse
are TRUE.
• Connect the conditional & its converse with the word “and”
• Write by joining the two parts of each conditional with the phrase “if
and only if” of “iff” for shorthand.
• Symbolically: p q
Bi-conditional Statements
Conditional Statement:
If two angles have the same measure, then the angles are congruent.
Converse:
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.
Conditional:
If x = 5, then x + 15 = 20.
Converse:
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.
Biconditional:
x = 5 if and only if x + 15 = 20.
Separate a Biconditional
• Write a biconditional as two conditionals that are converses of each
other.
Consider the biconditional statement:
A number is divisible by 3 if and only if the sum of its digits is divisible
by 3.
Statement 1:
If a number is divisible by 3, then the sum of its digits is divisible by 3.
Statement 2:
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.
Biconditional:
Lines are skew if and only if they are noncoplanar.
Conditional:
If lines are skew, then they are noncoplanar.
Converse:
If lines are noncoplanar, then they are skew.
Writing Definitions as Biconditionals
• Good Definitions:
 Help identify or classify an object
 Uses clearly understood terms
 Is precise avoiding words such as sort of and some
 Is reversible, meaning you can write a good definition as a biconditional (both
conditional and converse are true)
Show definition of perpendicular lines is reversible
Definition:
Perpendicular lines are two lines that intersect to form right angles
Conditional:
If two lines are perpendicular, then they intersect to form right angles.
Converse
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.”
Writing Definitions as Biconditionals
Show that the definition of triangle is reversible. Then write it as a
true biconditional.
Steps
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.
Conditional:
If a polygon is a triangle, then it has exactly three sides.
Converse:
If a polygon has exactly three sides, then it is a triangle.
Biconditional:
A polygon is a triangle if and only if it has exactly three sides.
Writing Definitions as Biconditionals
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.
Statement
Example
Symbolic
You read as
Conditional
If an angle is a straight angle,
then its measure is 180º.
p q
If p, then q.
Converse
If the measure of an angle is
180º, then it is a straight angle
q p
If q then p.
Inverse
If an angle is not a straight
angle, then its measure is not
180.
~p ~q
If not p, then
not q
Contrapositive
If an angle does not measure
180, then the angle is not a
straight angle.
~q ~p
If not q, then
not p.
Biconditional
An angle is a straight angle if
and only if its measure is 180º.
p q
p if and only if
q.
P iff q