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Converse of a Statement: Explanation and Example

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  • 0:01 If, Then Statements
  • 1:33 Converse
  • 2:24 Converse Examples
  • 4:22 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Just because a conditional statement is true, is the converse of the statement always going to be true? In this lesson, we'll learn the truth about the converse of statements.

If, Then Statements

There are certain conditional statements that you know are true. 'If I eat too many cookies, then I'm going to get fat.' I mean, that seems plausible, right?

But what about the opposite of that statement? 'If I get fat, then I ate too many cookies.' That's possibly true, but not necessarily true. Maybe I didn't eat any cookies. Maybe it was cake, pie, brownies or some other tasty, fatty food.

This opposite statement is a 'converse.' Wait, when did we start talking about shoes? No, this is a different kind of converse. This one refers to conditional statements, and it's what we're going to learn about here. I should note that this 'converse of a statement' is useful in geometry, not just when discussing my dietary choices.

Let's first talk about the two parts of a conditional statement. Here's a statement: 'If a polygon has three sides, then it's a triangle.' That first part, 'if a polygon has three sides,' is called the 'hypothesis.' It's stating something that may or may not be true. Then there's the second part, 'then it's a triangle.' This is a 'conclusion.' Whether or not the conclusion is true depends upon the truth of the hypothesis.

Here's a polygon:

polygon

Does it have three sides? Yes. So, our hypothesis is true. Therefore, our conclusion is true. It is a triangle.

Converse

If we reverse the hypothesis and conclusion, we have 'If a polygon is a triangle, then it has three sides.' This is called the converse of a statement. To get the converse, simply switch the hypothesis and conclusion. If we think of our original statement as 'if p, then q,' then the converse is 'if q, then p.' With our example, is the converse true?

Here's another triangle:

example polygon

So, the hypothesis, or first part, of our converse is true. Does it have three sides? Yes! So, the conclusion, or the second part, is true. Will it always be true?

In this statement, we can pretty easily tell that the converse of our statement is just as true as the original statement. Any time we have a triangle, it will have three sides.

Converse Examples

But that's not necessarily the case. Let's look at a few more examples. 'If we draw a median line in a triangle, then it bisects one side.' Okay, we can see that the median line below definitely bisects one side. So, our original statement is true. In fact, that's the definition of a median line - it bisects the side opposite the vertex from which it's drawn.

The original statement above is true.
median line bisecting triangle

What's the converse of our statement? 'If a line bisects one side of a triangle, then it's a median line.' In the triangle above, that's true. But what about with this other triangle?

The converse of the original statement is not true.
line bisecting triangle

Here, the line bisects one side of a triangle, so the hypothesis is true. But it's not a median line, is it? It doesn't extend from a vertex to the opposite side. So, in this example, the converse isn't true.

Let's try another. 'If a triangle has only two equal sides, then it is isosceles.' Okay, our hypothesis is on the lookout for a triangle with just two equal sides. Here's one:

The original statement is true.
isosceles triangle

This makes our hypothesis true. And is it isosceles? Yes! So, our conclusion is true. What's the converse of our statement? 'If a triangle is isosceles, then it has only two equal sides.' That sounds pretty good. But what about this triangle?

The conclusion of the converse is false.
equilateral triangle

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