Section 2 2 Biconditional Statements Biconditional statement a

Section 2 -2 Biconditional Statements

Biconditional statement • a statement that contains the phrase “if and only if”. • Equivalent to a conditional statement and its converse.

We can use iff to stand for “If and only if”

In order for a biconditional statement to be TRUE, both the conditional statement and its converse must be true.

Example #1: Write this biconditional statement as a conditional statement. • Two lines intersect if and only if their intersection is exactly one point.

Conditional Statement: • If two lines intersect, then their intersection is exactly one point. True

Now write the converse. • If their intersection is exactly one point, then two lines intersect. True

Example #2 Write this biconditional statement as a conditional statement. • Three lines are coplanar if and only if they lie in the same plane.

Conditional Statement: • If three lines are coplanar, then they lie in the same plane. True

Now write the converse. • If three lines lie in the same plane, then they are coplanar. True

Write the conditional as a biconditional statement. • If an angle is acute then it has a measure between 0° and 90°.

Write the converse • If an angle has a measure between 0° and 90°, then it is acute. True

Identify whether the converse is true or false • If it is true, then a biconditional can be written • If it is false, then a biconditional CAN NOT be written.

Bicondtional: • An angle is acute if and only if it has a measure between 0° and 90°.

Write the conditional as a biconditional statement. • If an animal is a leopard, then it has spots. Write the converse. • If an animal has spots, then is a leopard. False

Therefore a biconditional for this statement does not exist!

More Examples: Try It! Write each conditional as a biconditional statement, if possible. Be sure to give a counterexample if the converse is false!

1. If is perpendicular to , then their intersection forms a right angle. Converse: If, and intersect at a right angle, then they are perpendicular to each other. True

Biconditional: • is perpendicular to iff their intersection forms a right angle.

2. If 2 x < 49, then x < 7 Converse: 2 If x < 7, then x < 49. • Counterexample: let then Therefore, a biconditional can not be written!