Converse of a Theorem: Definition & Examples

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you will know how to find the converse of mathematical theorems. We will also discover how the converse of a theorem is not always true.

What Does Converse Mean?

In math, when you have a theorem, you likely have a converse theorem. Do you know what converse means? The converse of a theorem happens when the conclusion and hypothesis of a theorem are switched. For example, if you have a general theorem that says ''if this, then that'', then the converse theorem would say ''if that, then this''.

Here's another example:

  • Theorem: If you go to a Chinese restaurant, then you like Chinese food.
  • Converse Theorem: If you like Chinese food, then you go to a Chinese restaurant.

If you go to a Chinese restaurant, then you like Chinese food
converse theorem

Do you see how everything has been swapped? Additionally, the actual theorem can also be the converse of the converse theorem; they are converses of each other.

However, not all converses are true, even if the original statement is true. For example, the following statement is true all the time:

  • If it is raining, then my knee hurts.

However, the converse may not be true all the time:

  • If my knee hurts, then it is raining.

Your knee could hurt from other factors, but if you have a sensitivity to rain, your knee will hurt whenever it rains. The converse is not always true; this applies to mathematical theorems, also.

A Theorem

Let's look at this theory with a real-world mathematical theorem.

One famous theorem you've probably already worked with is called the Pythagorean Theorem. This theorem says:

  • If a triangle is a right triangle, then the square of the longest side of the triangle is equal to the sum of the squares of the other two sides.

You probably won't see the Pythagorean Theorem written out in word form; most likely you'll see it as a formula:

  • a2 + b2 = c2

The c stands for the hypotenuse of the right triangle (the longest side of the triangle), and the a and b stand for the other two sides of the triangle.

We know that this theorem holds true all the time.

The Converse

Now, let's look at the converse of the Pythagorean Theorem:

  • If the square of the longest side of the triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle

Does this hold true all the time? This key question is actually something that mathematicians have wondered and have successfully proven; the converse of the Pythagorean Theorem is always true. This means you can use the converse theorem to help prove a triangle is indeed a right triangle.

Examples

Another important theorem in math is the parallel lines theorem:

  • If two parallel lines are intersected by a transversal, then the alternate interior angles, alternate exterior angles, and the corresponding angles are congruent.

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