Login

The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Perpendicular Bisector Theorem: Proof and Example

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:01 The HL Theorem
  • 1:22 Proving the Theorem
  • 3:22 Practice Proof #1
  • 4:28 Practice Proof #2
  • 5:34 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Create an account to start this course today
Try it free for 5 days!
Create An Account

Recommended Lessons and Courses for You

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.

The HL Theorem

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.

Proving the Theorem

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

If AB=AD then the triangle is isosceles
Triangle ABD with bisector C

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?

Practice Proof #1

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!

To unlock this lesson you must be a Study.com Member.
Create your account

Register for a free trial

Are you a student or a teacher?
I am a teacher
What is your educational goal?
 Back

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back

Earning College Credit

Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it free for 5 days!
Create An Account
Support