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

Right triangles are aloof. They was standing apart from various other triangles, and also they get an exclusive set of congruence postulates and also theorems, choose the leg Acute Theorem and the leg Leg Theorem. The foot Acute Theorem seems to be absent "Angle," however "Leg Acute edge Theorem" is just too many words.

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Identifying residential property of right Triangles

Right triangles obtain their surname from one identify property:

It must, the course, it is in a triangle, an interpretation it is a three-sided polygon. That cannot have two internal right angles due to the fact that then it would certainly not be a triangle.

Right triangles can be any size, so lengthy as you get your forced three sides and three internal angles, among which should be 90°. The triangle deserve to face any type of direction. "Right" go not refer to direction; it comes from the Latin angulus rectus or "upright angle."

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

Right triangles have hypotenuses opposite their best angles. Hypotenuses space sides. The other two sides are referred to as legs, simply as an isosceles triangle has two legs.

Because all right triangles begin with one best angle, as soon as you shot to prove congruence, you have less work to do. Mathematicians constantly enjoy doing much less work.

Leg Acute (LA) Theorem

The Leg Acute Theorem, or LA Theorem, cannot take it its proud ar alongside the Los Angeles Rams, Los Angeles Angels, or Anaheim ducks (wait, what?). The LA organize has tiny to execute with The City of Angels.

If girlfriend recall ours freebie appropriate angle, you will automatically see how much time we have actually saved, due to the fact that we simply re-invented the Angle side Angle Postulate, cut out one angle, and made it distinct for best triangles.

Proving the LA Theorem

Below space two run-of-the-mill right triangles. Castle look like they are twins, but are they? We have labeled castle △WIT and also △FUN and also used hash marks to present that acute ∠W and acute ∠F room congruent.

We have additionally used hash clues (or ticks) to show sides IW ≅ UF. But, us have likewise used □ to determine their two appropriate angles, ∠I and ∠U.


Before you leap ahead to say, "Aha, The LA Theorem permits us come say the triangles are congruent," let"s do sure we have the right to really do that.

Right angles room congruent, since every appropriate angle will certainly measure 90°. Let"s testimonial what we have:

∠W ≅ ∠F (given)IW ≅ UF (given)∠I ≅ ∠U (right angles; deduced indigenous the symbol □, appropriate angle)

That, friend, is the Angle next Angle Postulate the congruent triangles. Come refresh your memory, the ASA Postulate claims two triangles are congruent if castle have equivalent congruent angles, corresponding contained sides, and another pair of matching angles.

We think we understand what you"re thinking: what if we had two different sides congruent, favor IT ≅ UN? What then?

Well, what of it? If you know ∠W ≅ ∠F room congruent, then you instantly know ∠T ≅ ∠N, because (and this is why best triangles space so cool) those 2 acute angles must include to 90°! If one pair of interior angles is congruent, the various other pair needs to be congruent, too! So friend still have actually Angle next Angeles -- er, Angle.

The organize is dubbed Leg Acute therefore you focus on acute legs, using those congruent appropriate angles as freebies, giving you 2 congruent angle to get Angle next Angle.

The leg Leg (LL) Theorem

But, friend, expect you have two right triangles that space not cooperating? You have two bag of matching congruent legs. That"s it. Castle refuse to cough up noþeles else. Space you walk to usage the foot Acute Theorem? Of course not! leaving it in your geometer"s toolbox and also take the end the sure-fire LL Theorem.

The Leg foot Theorem states Greg Legg play two periods with the Philadelphia Phillies -- nope; not correct Leg.

How have the right to this be?

Proving the LL Theorem

Here we have actually two right triangles, △BAT and also △GLV.


We have used ticks to show BA ≅ GL and AT ≅ LV. Do we understand anything else around these two triangles?

Sure! We recognize that ∠A ≅ ∠L because of the innocent-looking little right-angle square, □, in their inner angles. It may look like first, 2nd or third base, yet it is better than that.

What perform we have actually now?

BA ≅ GL (given)∠A ≅ ∠L (from □)AT ≅ LV (given)

What does the look like? That"s the Side Angle next Postulate, or SAS Postulate!

Practice proving Congruence

Let"s leaving the safety of spring training and shot our skills with part real major league games. Below is a rectangle, GRIN, through a diagonal line from inner right edge G to internal right edge I.


With just that one diagonal, we understand a significant amount about our polygon:

We produced two ideal triangles, △GRI and △GNIWe recognize ∠GRI ≅ ∠GNI (right angles of a rectangle)We understand ∠NGI ≅ ∠RIG (alternate internal angles of parallel currently intersected through a transversal, the diagonal)We understand the hypotenuses that both triangles space congruent (GI; reflex property)

With the hypotenuses and acute angles congruent, you obtain the HA Theorem, and they are congruent right triangles. The HA to organize is related to both these Theorems. Have the right to you watch why?

Like LA and LL, the HA Theorem provides the freebie appropriate angle to assist you and save girlfriend time!

Let"s shot Another Example

These two appropriate triangles hardly watch congruent.


Both their appropriate angles space at the reduced right corner, sure, however the ticks are reflecting congruent parts in various places!

That is since △LAF and also △PUN are not oriented the exact same way. See exactly how △LAF has the significant acute angle at the skinny top, while △PUN"s marked angle is way off to the narrow left? The congruent political parties seem to it is in in various places, too: AF ≅ PN.

To to compare these two best triangles, you must rotate and also reflect (flip) one of them. Then what perform you have?

The LA Theorem! castle have corresponding congruent legs and also acute angles; the two appropriate triangles space congruent.

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Lesson Summary

Now that you have operated through this lesson, you are able come recall and state the identifying building of appropriate triangles, state and apply the Leg Acute (LA) and Leg leg (LL) Theorems, and describe the relationship between the LA and also LL Theorems and also the Hypotenuse edge (HA) and Hypotenuse leg (HL) Theorems. With ideal triangles, you constantly get a "bonus" i can identify angle, the appropriate angle, in every congruence.

Next Lesson:

Isosceles Triangle Theorem

What you"ll learn:

After reviewing this text and also the multimedia, you will be maybe to:

Recall and also state the identifying property of ideal trianglesState and apply both the foot Acute (LA) and also Leg leg (LL) TheoremsDescribe the relationship between the LA and LL Theorems and the Hypotenuse angle (HA) and also Hypotenuse leg (HL) Theorems