Conditional Statements

“If-then”, or conditional statements, have two parts:

• hypothesis- if part
• conclusion- then part

In conditional statements, the hypothesis implies the conclusion. This is written as P→Q, and read as “P implies Q”.

Conditional statements have a truth value, meaning they are either true or false.

Example 1

True or False: If two planes intersect, then they are parallel.

Now obviously this is false; Intersecting planes are not parallel. Notice the two parts of the conditional statement:

• Hypothesis (P): two planes intersect

• Conclusion (Q): they are parallel

The Converse

The converse of a conditional statement is the reverse order of a conditional. In this case, Q→P, or Q implies P.

If the conditional is true, the converse is not necessarily true.

A counterexample is an example that proves the converse wrong.

Example 2

Identify P and Q, determine truth value. Then write converse and determine truth value. If the converse is false, write a counterexample.

If two lines intersect, then they aren’t parallel.

Let’s break it down into the P and Q.

• P: two lines intersect
• Q: they aren’t parallel

It’s true. Let’s write the converse by switching the P and Q.

Converse: If two lines are parallel, then they don’t intersect.

This may seem true, but skew lines also don’t intersect, so it must be false. The skew lines will be our counterexample.

Counterexample: Skew lines

Further Reading

Geometry Help: Points, Lines, and Planes

Geometry Help: Segments, Rays, Parallel Lines, and Planes

Geometry Help: Measuring Segments and Angles

Geometry Help: Biconditionals and Definitions