Conditional Statements and Their Converse

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Conditional and converse statements

Geometry is a wonderful part of mathematics for people who don't like a lot of numbers. It has shapes and angles, and it also has logic. Logic is formal, correct thinking, reasoning, and inference. Logic is not something humans are born with; we have to learn it, and geometry is a great way to learn to be logical.

Converse statements

You may know the word converse for a verb meaning to chat, or for a noun as a particular brand of footwear. Neither of those is how mathematicians use converse. Converse and inverse are connected concepts in making conditional statements.

To create the converse of a conditional statement, switch the hypothesis and conclusion. To create the inverse of a conditional statement, turn both hypothesis and conclusion to the negative.

Converse statement examples

• If I eat a pint of ice cream, then I will gain weight. (Conditional Statement)

• If I gained weight, then I ate a pint of ice cream. (Converse)

• If I do not eat a pint of ice cream, then I will not gain weight. (Inverse)

Conditional statements

Conditional statements set up conditions that could be true or false. These conditions lead to a result that may or may not be true. Conditional statements start with a hypothesis and end with a conclusion.

Conditional statement examples

• If my cat is hungry, then she will rub my leg.

• If a polygon has exactly four sides, then it is a quadrilateral.

• If triangles are congruent, then they have equal corresponding angles.

Here are examples of conditional statements with false hypotheses:

• If I am 9 meters tall, then I can play basketball.

• If a square has three sides, then its interior angles add to 180°.

You can test the hypothesis immediately: Are you 9 meters tall? Do squares have three sides?

These conditional statements result in false conclusions because they started with false hypotheses.

Creating conditional statements

Conditional statements begin with "If" to introduce the hypothesis. The hypothesis is the part that sets up the condition leading to a conclusion. The conclusion begins with "then," like this:

• If my dog barks, then my dog observed something that excited him.

You will see conditional statements in geometry all the time. You can set up your own conditional statements. Here is one for an isosceles triangle:

• If the triangle is isosceles, then only two of its sides are equal in length.

Exchanging parts of conditional statements

You can switch the hypothesis and conclusion of a conditional statement. You take the conclusion and make it the beginning, and take the hypothesis and make it the end:

• If my dog observes something that excites him, then he barks.

• If triangles have equal corresponding sides, then they are congruent.

Converse of a conditional statement

The converse of a true conditional statement does not automatically produce another true statement. It might create a true statement, or it could create nonsense:

• If a polygon is a square, then it is also a quadrilateral.

That statement is true. But the converse of that is nonsense:

• If a polygon is a quadrilateral, then it is also a square.

We know it is untrue because plenty of quadrilaterals exist that are not squares.

Geometry and conditional statements

Many times in geometry we see postulates and theorems that seem like they could become conditional statements and converse conditional statements:

• Parallel lines never meet. (Postulate)

• If two lines are parallel, then they are lines that never meet. (Conditional Statement)

• If two lines never meet, then they are parallel. (Converse)

Example #2

• Adjacent angles share a common side. (Postulate)

• If angles share a common side, then they are adjacent. (Conditional Statement)

• If angles are adjacent, then they share a common side. (Converse)

Some postulates are even written as conditional statements:

• If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

• If two points lie in a plane, then the line joining them lies in that plane.

Practice conditional statements

Below we have equilateral triangle △NAP. We can set up conditional statements about it. Here are five statements. Decide which ones are conditional, which are not conditional, and which conditional statements are true:

1. If △NAP is equilateral, then its interior angles are all equal.

2. If △NAP is equilateral, then interior ∠N is 60°.

3. If interior ∠N is 60°, then △NAP is equilateral.

4. Equilateral triangles have equal interior angles.

5. If △NAP is equilateral, then it is also isosceles.

Statements 1, 2, and 5 are all true conditional statements (If … then).

Statement 3 is a converse of statement 2.

Statement 4 is not a conditional statement, but it is true. You have enough information to change statement 4 into a conditional statement.

Let's check the converse statement, 3, to see if it is true. Can you create a triangle with one interior angle measuring 60° but with the other angles having different measures?

Of course you can, like a 30−60−90 triangle, which is definitely not equilateral. So the converse statement is not true.

Lesson summary

In this lesson you have learned to identify and explain conditional statements and create your own conditional statements. You know conditional statements could be true or false. You are able to exchange the hypothesis and conclusion of a conditional statement to produce a converse of the statement, and you can test to see if the converse of a true conditional statement is true.