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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.

In this lesson, we'll learn about the hypotenuse leg theorem. This theorem enables us to prove two right triangles are congruent based on just two sides.

Right triangles are the best. They're always trying to help us out. In the real world, they're the cheese that complements the cracker, the wedge that keeps doors from closing, and the toast when it's properly cut and, you know, not just hacked down the middle like we're Neanderthals or something.

In geometry, right triangles are our friends, too. In a right triangle, we always know one of the angles is 90 degrees. And knowing is half the battle. Well, since there are three angles, I guess it's just a third of the battle. But that's still a lot of the battle.

We also have awesome tools, like the Pythagorean theorem, or *a*^2 + *b*^2 = *c*^2. This allows us to always figure out the third side of a triangle if we know two.

And then there's the **hypotenuse leg theorem**, or **HL theorem**. This theorem states that 'if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.'

This is kind of like the SAS, or side-angle-side postulate. But SAS requires you to know two sides and the included angle. With the HL theorem, you know two sides and an angle, but the angle you know is the right angle, which isn't the included angle between the hypotenuse and a leg.

There are several different ways we can verify that this theorem checks out. One simple way is with this triangle.

Here, we're told that *AB* = *AD*. So *ABD* is an isosceles triangle. Isosceles triangles are good and all, but can you cut your toast into an isosceles triangle? Well, I guess you could, but would you want to?

Anyway, we also know that *AC* is an altitude line. That means it's perpendicular to *BD*. Perpendicular lines form right angles, so angles *ACB* and *ACD* are right angles. That makes our two smaller triangles, *ABC* and *ADC*, right triangles. Hooray! Not just one friend, but two.

What are the hypotenuses of these right triangles? *AB* and *AD*, and we know they are equal to each other. Plus, we know that *AC* = *AC* because, well, they're the same line. More formally, we call this the reflexive property. So *AB* and *AC* are equal to *AD* and *AC*. That's a hypotenuse and a leg pair in two right triangles, which is the definition of the HL theorem.

If this theorem is correct, then these must be congruent triangles. Can we be sure? Well, we know angles *B* and *D* are equal. They're the sides opposite the equal sides of isosceles triangle *ABD*.

We also know that angles *BAC* and *DAC* are equal. Why? Because this altitude line in an isosceles triangle bisects the angle. It also bisects *BD*, which makes *BC* equal to *CD*.

We just showed that all three angles and all three sides of our two right triangles are congruent. That's the definition of congruent triangles. Therefore, we just verified the HL theorem. I guess we knew it would work out. Right triangles don't let us down, right?

What about seeing this theorem in action? Let's try a proof with these two triangles. We're given that angles *O* and *X* are right angles. Also, *MN* is congruent to *ZY* and *NO* is congruent to *YX*. Can we prove that angle *M* is congruent to angle *Z*?

First, let's state that angles *O* and *X* are right angles. We're given that. That means triangles *MNO* and *ZYX* are right triangles. They have one right angle, and that's the definition of right triangles.

Let's also state that we're given that *MN* is congruent to *ZY* and *NO* is congruent to *YX*. Okay, right triangles and congruent hypotenuses and congruent legs. Now we can state that triangle *MNO* is congruent to triangle *ZYX* using the HL theorem.

Finally, we can state that angle *M* is congruent to angle *Z* because corresponding parts of congruent triangles are congruent, or CPCTC. And that's it!

How about another? Here's a butterfly. It's also two connected triangles. We're given that angle *PRQ* is a right angle. Plus, we know that *PQ* is congruent to *TS* and *PR* is congruent to *TR*. Can we prove that *QR* is congruent to *SR*?

Let's start by stating that angle *PRQ* is a right angle. That's given. That also means that angle *SRT* is a right angle because they're vertical angles. So now we can state that triangles *PRQ* and *TRS* are right triangles. That's the definition of right triangles.

We're a third of the way there. And that's half the battle. Or, well, you know, a third. But wait, what else were we given? *PQ* is congruent to *TS*. *PQ* and *TS*? Hypotenuse and hypotenuse! And *PR* is congruent to *TR*. Those are legs.

So triangle *PRQ* is congruent to triangle *TRS* because of the HL theorem. Now we can wrap this up by stating that *QR* is congruent to *SR* because of CPCTC again.

In summary, we learned about the **hypotenuse leg**, or **HL**, **theorem**. This tells us that if one leg and the hypotenuse of one right triangle are congruent to one leg and the hypotenuse of another right triangle, then the triangles are congruent. This is one of the many ways in which right triangles are our friends.

It's essentially a modified version of the SAS, or side-angle-side postulate. Since it's a right triangle, we don't need the included angle, just the right angle. Once we've determined triangles are congruent using the HL theorem, we know that all three corresponding sides and angles are congruent.

At the end of this lesson, you should be able to state the hypotenuse leg theorem of congruent triangles and use it to prove congruency.

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Geometry: High School15 chapters | 160 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:16
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- Similarity Transformations in Corresponding Figures 7:28
- 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
- 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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