Biconditional and Conditional Statements

Biconditional Statements

Example 1

Conditional: If two lines are perpendicular, then they form a right angle.

Converse: If two lines form a right angle, then they are perpendicular.

Both the conditional and converse are true, so this is biconditional.

Example 1 Continued

Now we can write the statement in the "if and only if" form.

Biconditional: Two lines are perpendicular if and only if they form a right angle.

*Notice how the words "if" and "then" were removed and the hypothesis and conclusion were connected with the phrase "if and only if" in the middle.

Example 2

Conditional: If a triangle is a right triangle, then it has two acute angles.

Converse: If a triangle has two acute angles, then it is a right triangle.

The conditional is true but the converse is false. So this is not a biconditional statement and it would not make sense to write this in the "if and only if" form.

Open Ended

Write the converse of the statement and if both are true write as a biconditional.

Conditional: If x is even, then 2x is even.

Answer

Converse: If 2x is even, then x is even. False

Example: If 2x=6, then x=3. 3 is odd.

Not Biconditonal

Open Ended

Write the converse of the statement and if both are true write as a biconditional.

Conditional: If a polygon is a hexagon, then it has six sides.

Answer

Converse: If a polygon has six sides, then it is a hexagon. True

Biconditonal: A polygon is a hexagon if and only if it has six sides.

OR: A polgyon has six sides if and only if it is a hexagon.

The biconditional can be written in either order since they are both true.

Good Definition

A good definition should be biconditional. Such as the last example about hexagons.

Examples

Two different lines are parallel if and only if they have the same slope. This is a good definition of parallel lines because it is biconditional.

Two angles are a linear pair if and only if the two angles are supplementary. This is a bad definition of linear pair because it is not biconditional. Supplementary angles don't have to be adjacent so they don't have to be a linear pair.

Multiple Choice

Determine if the following is a good or bad definition:

An equation is linear if and only if its graph forms a straight line.

• good definition

• bad definition

Multiple Choice

Determine if the following is a good or bad definition:

The converse of a conditional statement is true if and only if the contrapositive is true.

• good definition

• bad definition

Multiple Select

Select the if-then statements that could have formed the following biconditional.

Two lines are perpendicular if and only if their slopes are opposite reciprocals.

• If two lines are perpendicular, then their slopes are the same.

• If two lines are perpendicular, then their slopes are opposite reciprocals.

• If two lines have opposite reciprocal slopes, then they are perpendicular.

• If two lines have opposite reciprocal slopes, then they are parallel.

• Two lines will always be parallel or perpendicular.