Back To CourseGeometry: High School
15 chapters | 160 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
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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Back To CourseGeometry: High School
15 chapters | 160 lessons