What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive.

But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson.

Given an if-then statement “if pp , then qq ,” we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they cancel school.”

*“It rains” *is the hypothesis.

*“They cancel school” *is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.

The converse of *“If it rains, then they cancel school” *is *“If they cancel school, then it rains.”*

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

The inverse of *“If it rains, then they cancel school” *is *“If it does not rain, then they do not cancel school.”*

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

The contrapositive of *“If it rains, then they cancel school” *is *“If they do not cancel school, then it does not rain.”*

Statement | If pp , then qq . |

Converse | If qq , then pp . |

Inverse | If not pp , then not qq . |

Contrapositive | If not qq , then not pp . |

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

**Example 1:**

Statement | If two angles are congruent, then they have the same measure. |

Converse | If two angles have the same measure, then they are congruent. |

Inverse | If two angles are not congruent, then they do not have the same measure. |

Contrapositive | If two angles do not have the same measure, then they are not congruent. |

In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!

**Example 2:**

Statement | If a quadrilateral is a rectangle, then it has two pairs of parallel sides. |

Converse | If a quadrilateral has two pairs of parallel sides, then it is a rectangle. (FALSE!) |

Inverse | If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. (FALSE!) |

Contrapositive | If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle. |