2-3 Biconditionals and
Definitions
Biconditional
• A biconditional is a single true statement that combines a true
conditional and its true converse.
• You can write a biconditional by joining the two parts of each
conditional with the phrase “if and only if.”
• You can write a biconditional as two conditionals that are converses.
• A biconditional combines pq and qp as p  q.
• This is read “p if and only if q.”
• It means “If p, then q, and if q, then p.”
Definitions
• A definition is good if it can be written as a biconditional.
• When a conditional and its converse is true, it is said to be
“reversible.” Therefore, a test to see if a definition is good is to see if
it is reversible.
Examples (begin on p.101)
Directions: Each conditional statement below is true. Write its
converse. If the converse is also true, combine the statements as a
biconditional.
7. If two segments have the same length, then they are congruent.
• Converse: If two segments are congruent, then they have the same
length.
• Biconditional:
Two segments have the same length if and only if they are congruent.
Directions: Write the two statements that form each biconditional.
13. A line bisects a segment if and only if the line intersects the
segment only at its midpoint.
If a line bisects a segment, then it intersects the segment only at its
midpoint.
If a line intersects a segment only at its midpoint, then the line bisects
the segment.
Directions: Test each statement to see if it is reversible. If so, write it
as a true biconditional. If not, write not reversible.
20. Complementary angles are two angles with measures that have a
sum of 90.
Two angles are complementary if and only if the measures of the
angles have a sum of 90.
Directions: Is each statement below a good definition? If not,
explain.
26. A segment is a part of a line.
No, it is not precise; a point or a ray can be a part of a line.
Directions: Write each statement as a biconditional.
35. Points in Quadrant III have two negative coordinates.
A point is in Quadrant III if and only if it has two negative coordinates.