Congruency of Right Triangles — LA & LL Theorems

Malcolm McKinsey
Written by
Malcolm McKinsey
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Paul Mazzola

Leg Acute (LA) and Leg Leg (LL) Theorems

Right triangles are aloof. They stand apart from other triangles, and they get an exclusive set of congruence postulates and theorems, like the Leg Acute Theorem and the Leg Leg Theorem. The Leg Acute Theorem seems to be missing "Angle," but "Leg Acute Angle Theorem" is just too many words.

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Identifying property of right triangles

Right triangles get their name from one identifying property:

It must, of course, be a triangle, meaning it is a three-sided polygon. It cannot have two interior right angles because then it would not be a triangle.

Right triangles can be any size, so long as you get your required three sides and three interior angles, one of which must be 90°. The triangle can face any direction. "Right" does not refer to direction; it comes from the Latin angulus rectus or "upright angle."

Notice the elegance of the unspoken consequence of one right angle: the other two angles of a right triangle must each be acute, or less than 90° each. In fact, they will be complementary, meaning they will add to 90° (not free as in complimentary peanuts).

Right triangles have hypotenuses opposite their right angles. Hypotenuses are sides. The other two sides are called legs, just as an isosceles triangle has two legs.

Because all right triangles start with one right angle, when you try to prove congruence, you have less work to do. Mathematicians always enjoy doing less work.

Leg acute (LA) theorem

The Leg Acute Theorem, or LA Theorem, cannot take its proud place alongside the Los Angeles Rams, Los Angeles Angels, or Anaheim Ducks (wait, what?). The LA Theorem has little to do with The City of Angels.

If you recall our freebie right angle, you will immediately see how much time we have saved, because we just re-invented the Angle Side Angle Postulate, cut out an angle, and made it special for right triangles.

Proving the LA theorem

Below are two run-of-the-mill right triangles. They look like they are twins, but are they? We have labeled them WIT\triangle WIT and FUN\triangle FUN and used hash marks to show that acute W\angle W and acute F\angle F are congruent.

We have also used hash marks (or ticks) to show sides IWUFIW\cong UF. But, we have also used □ to identify their two right angles, I\angle I and U\angle U.

 LA theorem
LA theorem

Before you leap ahead to say, "Aha, The LA Theorem allows us to say the triangles are congruent," let's make sure we can really do that.

Right angles are congruent, since every right angle will measure 90°. Let's review what we have:

  • WF\angle W\cong \angle F (given)

  • IWUFIW\cong UF (given)

  • IU\angle I\cong \angle U (right angles; deduced from the symbol □, right angle)

That, friend, is the Angle Side Angle Postulate of congruent triangles. To refresh your memory, the ASA Postulate says two triangles are congruent if they have corresponding congruent angles, corresponding included sides, and another pair of corresponding angles.

We think we know what you're thinking: what if we had two different sides congruent, like ITUNIT\cong UN? What then?

Well, what of it? If you know WF\angle W\cong \angle F are congruent, then you automatically know TN\angle T\cong \angle N, because (and this is why right triangles are so cool) those two acute angles must add to 90°! If one pair of interior angles is congruent, the other pair has to be congruent, too! So you still have Angle Side Angle.

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The theorem is called Leg Acute so you focus on acute legs, using those congruent right angles as freebies, giving you two congruent angles to get Angle Side Angle.

The leg leg (LL) theorem

But, friend, suppose you have two right triangles that are not cooperating? You have two pairs of corresponding congruent legs. That's it. They refuse to cough up anything else. Are you going to use the Leg Acute Theorem? Of course not! Leave it in your geometer's toolbox and take out the sure-fire LL Theorem.

The Leg Leg Theorem says Greg Legg played two seasons with the Philadelphia Phillies -- nope; wrong Leg.

How can this be?

Proving the LL theorem

Here we have two right triangles, BAT\triangle BAT and GLV\triangle GLV.

 LL theorem
LL theorem

We have used ticks to show BAGLBA\cong GL and ATLVAT\cong LV. Do we know anything else about these two triangles?

Sure! We know that AL\angle A\cong \angle L because of that innocent-looking little right-angle square, □, in their interior angles. It may look like first, second or third base, but it is better than that.

What do we have now?

  1. BAGLBA\cong GL (given)

  2. AL\angle A\cong \angle L (from □)

  3. ATLVAT\cong LV (given)

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What does that look like? That's the Side Angle Side Postulate, or SAS Postulate!

Practice proving congruence

Let's leave the safety of spring training and try our skills with some real major league games. Here is a rectangle, GRIN, with a diagonal from interior right angle G to interior right angle I.

Congruent triangles
Congruent triangles

With just that one diagonal, we know a tremendous amount about our polygon:

  1. We created two right triangles, GRI\triangle GRI and GNI\triangle GNI

  2. We know GRIGNI\angle GRI\cong \angle GNI (right angles of a rectangle)

  3. We know NGIRIG\angle NGI\cong \angle RIG alternate interior angles of parallel lines intersected by a transversal, the diagonal)

  4. We know the hypotenuses of both triangles are congruent (GI; reflexive property)

With the hypotenuses and acute angles congruent, you get the HA Theorem, and they are congruent right triangles. The HA Theorem is related to both these Theorems. Can you see why?

Like LA and LL, the HA Theorem uses the freebie right angle to help you and save you time!

Let's try another example. These two right triangles hardly look congruent.

Congruent right triangles
Congruent right triangles

Both their right angles are at the lower right corner, sure, but the ticks are showing congruent parts in different places!

That is because LAF\triangle LAF and PUN\triangle PUN are not oriented the same way. See how LAF\triangle LAF has the marked acute angle at the skinny top, while PUN\triangle PUN's marked angle is way off to the narrow left? The congruent sides seem to be in different places, too: AFPNAF\cong PN.

To compare these two right triangles, you must rotate and reflect (flip) one of them. Then what do you have?

The LA Theorem! They have corresponding congruent legs and acute angles; the two right triangles are congruent.

Lesson summary

Now that you have worked through this lesson, you are able to recall and state the identifying property of right triangles, state and apply the Leg Acute (LA) and Leg Leg (LL) Theorems, and describe the relationship between the LA and LL Theorems and the Hypotenuse Angle (HA) and Hypotenuse Leg (HL) Theorems. With right triangles, you always get a "bonus" identifiable angle, the right angle, in every congruence.