Biconditional

Conditional Statements

Example 1

Conditional: If two angles are adjacent, then they share a vertex and side.

Hypothesis: two angles are adjacent

Conclusion: they share a vertex and side

This is a true conditional statement because when we have adjacent angles, they do share a vertex and side.

Multiple Choice

If two angles are vertical angles, then they share a vertex.

What is the hypothesis?

• two angles are vertical angles

• they share a vertex

Multiple Choice

If two angles are vertical angles, then they share a vertex.

What is the conclusion?

• two angles are vertical angles

• they share a vertex

Multiple Choice

If two angles are vertical angles, then they share a vertex.

True or False?

• True

• False

Converse Statement

When writing the converse of a statement, you switch the hypothesis and conclusion.

Note: If the conditional and converse are true, then the statement is called biconditional. It can then be written in the "if and only if" form.

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.

Example 3

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.

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

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.

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

Good Definition

A good definition should be biconditional. Such as the 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 conditional is true.

• good definition

• bad definition