Biconditional Statements & Definitions
section 2-4
When you combine a conditional statement and its converse into a single
statement, you create a biconditional statement.
example: The conditional statement "If a triangle has 3 congruent sides, then it is
equilateral." can be combined with its converse "If it is equilateral, then
a triangle has 3 congruent sides." into a single equivalent statement:
"A triangle has 3 congruent sides if and only if it is equilateral."
A biconditional statement is a statement that can be written in the form:
“p if and only if q.” This means “if p, then q” and “if q, then p.”
The biconditional “p if and only if q” can also be written as “p iff q” or p
p
q means p
q and q
q.
p
problem # 1 - Write the conditional statement and converse within each biconditional.
a. Two angles are congruent if and only if their measures are equal.
b. Barbara is a club member if and only if she has paid the $20 dues.
c. 3x - 6 = 9 if and only if x = 5.
problem # 2 - For each conditional, write its converse and a biconditional statement.
a. If a point is a midpoint, then it divides the segment into two
congruent segments.
b. If the date is December 25th, then it is Christmas Day.
For a biconditional statement to be true, both the conditional statement and its
converse must be true. If either the conditional or the converse is false, then
the biconditional statement is false.
problem # 3 - Determine whether each biconditional is true. If false, give a
counterexample.
a. A square has a side length of 5 if and only if it has an area of 25.
b. x = 4
x 2 = 16
c. An angle is a right angle iff its measure is 90 .
d. A rectangle has side lengths of 12 cm and 25 cm if and only if
its area is 300 cm2.
In geometry, biconditional statements are used to write definitions.
definition - a statement that describes a mathematical object and can be
written as a true biconditional.
In the glossary at the back of your textbook, a polygon is defined as a closed
plane figure formed by three or more segments such that each segment
intersects exactly two other segments only at their endpoints, and no two
segments with a common endpoint are collinear.
A triangle is defined as a three-sided polygon, and a quadrilateral is a foursided polygon.
Think of definitions as being reversible. A good definition can be used both
forward and backward. To make certain that a definition is precise, it helps to
write it as a biconditional statement. Postulates, on the other hand, are not
necessarily true when reversed.
problem # 4 - Write each definition as a biconditional statement.
a. A quadrilateral is a four-sided polygon.
b. The measure of a straight angle is 180 .
c. A right angle measures 90 .