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

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  • 0:01 The HL Theorem
  • 1:22 Proving the Theorem
  • 3:22 Practice Proof #1
  • 4:28 Practice Proof #2
  • 5:34 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.

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?

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