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High School Geometry: Tutoring Solution14 chapters | 161 lessons

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.

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.

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.

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:

*a*2 +*b*2 =*c*2

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.

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.

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.

This theorem is useful when you need to find the measures of various angles formed from the transversal of two parallel lines because it tells you which angles are congruent (or equal) to each other. It can actually be split into several converse theorems.

These converse theorems are outside the scope of this lesson and are used in determining whether or not two lines are parallel. Know that they are very useful, and have been proven over time by various mathemeticians:

- If two alternate interior angles are congruent, then the two lines cut by a transversal are parallel.
- If two alternate exterior angles are congruent, then the two lines cut by a transversal are parallel.
- If two corresponding angles are congruent, then the two lines cut by a transversal are parallel.

Let's review!

The **converse** of a theorem is when the conclusion and hypothesis are switched.

For example, when the theorem states:

- ''If this, then that''

Then the converse of the theorem is:

- ''If that, then this''

The conclusion has been replaced with the hypothesis and vice versa. Many true mathematical theorems have very useful converses that have also been proven to be true. Examples include the Pythagorean and Parallel Theorems.

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21 in chapter 5 of the course:

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High School Geometry: Tutoring Solution14 chapters | 161 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
- 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
- Angle of Elevation: Definition, Formula & Examples 4:50
- Centroid: Definition, Theorem & Formula 6:15
- Flow Proof in Geometry: Definition & Examples 4:31
- Infinite Sequence: Definition & Examples 6:39
- Law of Detachment in Geometry: Definition & Examples 4:30
- Phase Shift: Definition & Formula
- Side-Side-Side (SSS) Triangle: Formula & Theorem
- Square Pyramid: Definition & Properties 3:59
- What is an Acute Triangle? - Definition, Facts & Example 2:44
- Converse of a Theorem: Definition & Examples
- Go to Triangles, Theorems and Proofs: Tutoring Solution

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