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Geometry: High School15 chapters | 160 lessons

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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.

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:

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

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:

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.

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.

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?

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:

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?

We know this triangle is equilateral because it has three equal sides. But is it also isosceles? It is. An isosceles triangle, by definition, has at least two equal sides, so all equilateral triangles are also isosceles. That means that this triangle is isosceles, making the hypothesis of our converse true. But does it have only two equal sides? No. So, the conclusion is false.

Again, this is an example of why you can't assume that the converse of a statement is true.

In summary, we learned about conditional statements. These are if, then statements, such as 'If I eat too many cookies, then I'm going to get fat.' The first part, 'if I eat too many cookies,' is called the 'hypothesis.' The second part, 'then I'm going to get fat,' is the conclusion.

If we swap the hypothesis and conclusion, we get 'If I get fat, then I ate too many cookies.' This is called the **converse**. We always need to verify the converse of a conditional statement. We can't assume it's true just because the original statement is true. This is the case whether we're dealing with triangles or cookies. Remember, there are always brownies, pie, cake and so many other tasty desserts.

Once you've finished studying the contents of this lesson, you may have the ability to:

- Explain what conditional statements are and provide examples
- Switch the hypothesis and conclusion of a conditional statement
- Follow the steps necessary to get the converse of a conditional statement

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Geometry: High School15 chapters | 160 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- How to Prove Relationships in Figures using Congruence & Similarity 5:14
- Practice Proving Relationships using Congruence & Similarity 6:16
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
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