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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
If you're a triangle, finding out that you're congruent to another triangle is a big deal. Imagine finding out one day that you have a twin that you didn't know about. How amazing would that be? It's like having a spare 'you' suddenly enter your life.
In geometry, we try to find triangle twins in any way we can. There are all kinds of methods, like side-side-side, angle-side-angle, side-angle-side and more. In the real world, it doesn't work that way. You can't just compare legs with a stranger to test for congruency.
With two right triangles, we already know that they have something in common - those right angles. So, it's like they're at least cousins. And we can prove they're congruent with the hypotenuse angle theorem.
The hypotenuse angle theorem, also known as the HA theorem, states that 'if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.'
How can we verify congruency with just a hypotenuse and an acute angle? It's like saying two people are twins because they have the same height and hair color. Let's look at a couple of triangles.
Here's triangle ABC:
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Now along comes triangle XYZ:
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Could they be twins? They're both right triangles. Angles B and Y are each 90 degrees. We're told that AC is congruent to XZ. So, that's one hypotenuse that's congruent to the other. And we're told that angle A is congruent to angle X. That's good, but it's not like a DNA test.
Or is it? In triangle ABC, what's the sum of the interior angles? 180. What about with triangle XYZ? It's also 180. So, if two angles are congruent, like A and X, and another two angles are congruent, like B and Y, then the other angles, C and Z, must also be congruent.
So, now we have angle A, side AC and angle C congruent to angle X, side XZ and angle Z. And that's angle-side-angle, or ASA. That means that the HA theorem is really just a simplification or variation of the ASA postulate that works with right triangles.
Let's try to find some twins with some proofs. You know, you're not twins without proof. Here are two triangles:
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They're very close. Together, they look kinda like a kite, don't they? Maybe they like to fly kites together. But are they just really good friends, or are they twins?
We're given that angles R and S are right angles. And we're also given that angle SQT is congruent to angle RQT. Let's say we want to determine if RT is congruent to ST.
Let's start our proof by collecting DNA samples from each triangle. Wait, what? Two-dimensional polygons don't have DNA? Oh. Then I guess we'll need to do an ordinary proof. Okay, first, we know that angles R and S are right angles. We're given that. That means that triangles QST and QRT are right triangles. That's the definition of a right triangle.
Next, we know that angle SQT is congruent to angle RQT. That's given. And we know that QT is congruent to QT because of the reflexive property. Now we can say that triangle QST is congruent to QRT because of the HA theorem. So, they're not just kite buddies; they're twins!
That enables us to say that RT is congruent to ST due to CPCTC, or corresponding parts of congruent triangles are congruent. And we're done!
How about one more? Here are two triangles that are also close:
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How close? They're practically joined at the vertex. Oh, triangle humor.
Anyway, we're given that AC is congruent to CE and that angles B and D are right angles. We want to know if AB is congruent to DE. First, we'll need to determine if the triangles are congruent.
Let's start by stating that angle B is a right angle. Next, angle D is a right angle. Okay, so ABC and CDE are right triangles. One right angle apiece and that's the definition of right triangles.
Now let's state that AC is congruent to CE. That's given. So, right triangles, and we know one hypotenuse is congruent to the other. That's not enough, is it? But wait. We can say that angle ACB is congruent to angle DCE. Why? They're vertical angles, and vertical angles are congruent.
Now it's time to bust out our HA theorem and state that triangles ABD and CDE are congruent. So, they are like conjoined twins! Now we can finish our proof by using CPCTC to state that AB is congruent to DE.
In summary, we learned a valuable lesson about twins. Well, maybe not human twins. But we did learn about right triangle twins. Specifically, we focused on the hypotenuse angle theorem, or the HA theorem. This theorem states that 'if the hypotenuse and one acute angle of a right triangle are congruent to the hypotenuse and one acute angle of another right triangle, then the triangles are congruent.' We saw how this is really just a variation of ASA, or angle-side-angle. We then used this theorem in a pair of proofs to help us demonstrate congruency. And all this without any DNA tests!
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