Right Triangle Congruence Theorem

Introduction To Right Triangle Congruence Theorems

Besides, equilateral and isosceles triangles having special characteristics, Right triangles are also quite crucial in the learning of geometry. For two right triangles that measure the same in shape and size of the corresponding sides as well as measure the same of the corresponding angles are called congruent right triangles. However, before proceeding to congruence theorem, it is important to understand the properties of Right Triangles beforehand.

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Properties of Right Triangles

Know that Right triangles are somewhat peculiar in characteristic and aren't like other, typical triangles.Typical triangles only have 3 sides and 3 angles which can be long, short, wide or any random measure. There's no order or uniformity. However right angled triangles are different in a way:-

  • Right triangles are uniform with a clean and tidy right angle.

  • Right triangles have a hypotenuse which is always the longest side, and always in the same position, opposite the 90 degree angle.

  • Right triangles have the legs that are the other two sides which meet to form a 90-degree interior angle

  • While other triangles require three matches like the side-angle-side hypothesize amongst others to prove congruency, right triangles only need leg, angle postulate.

Now being mindful of all the properties of right triangles, let’s take a quick rundown on how to easily prove the congruence of right triangles using congruence theorems.


Congruence Theorems To Prove Two Right Triangles Are Congruent

In the chapter, you will study two theorems that will help prove when the two right triangles are in congruence to one another. These two congruence theorem are very useful shortcuts for proving similarity of two right triangles that include;-

  1. The LA Theorem (leg-acute theorem),

  2. The LL Theorem (leg-leg)

 

  1. The LA Theorem

Do not confuse it with Los Angeles. It’s the leg-acute theorem of congruence that denotes if the leg and an acute angle of one right triangle measures similar to the corresponding leg and acute angle of another right triangle, then the triangles are in congruence to one another.

If you recall the giveaway right angle, you will instantly realize the amount of time we have saved, because we just re-modeled the Angle Side Angle (ASA) congruence rule, snipped off an angle, and made it extra special for right triangles.


Proving the LA Theorem

Let's take a look at two Example triangles, ABC and DEF.

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The above two congruent right triangles ABC and DEF surely look like they belong in a marching trumpet player together, don't they? You know that they're both right triangles. Both Angles B and E are 90 degrees each. And you know AB measures the same to DE and angle A is congruent to angle D.

So, Using the LA theorem, we've got a leg and an acute angle that match, so they're congruent.' But how is this true?

Observe, since B and E are congruent, too, that this is really like the ASA rule. The fact that they're right triangles just provides us a shortcut. And even if we have not had included sides, AB and DE here, it would still be like ASA.

What if we know A and D are similar, but then what about BC and EF? Well, since the total of the angles of a triangle is 180 degrees, we know that C and F, too, shall be congruent to each other. So we still get our ASA postulate.


  1. The LL Theorem

Again, do not confuse it with LandLine. The LL theorem is the leg-leg theorem which states that if the length of the legs of one right triangle measures similar to the legs of another right triangle, then the triangles are congruent to one another. They definitely look like they belong in a marching band with matching pants, don't they? Their legs reflect mirror image, right?


Proving the LL Theorem

Let's take a look at two Example triangles, MNO and XYZ

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The above two congruent right triangles MNO and XYZ seem as if triangle MNO plays the aerophone while XYZ plays the metallophone. We are well familiar, they're right triangles. Both Angles N and Y are 90 degrees. If we are aware that MN is congruent to XY and NO is congruent to YZ, then we have got the two legs. This immediately allows us to say they're congruent to each other based upon the LL theorem.

Observe, The LL theorem is really like the SAS rule. If you recall that the legs of a right triangle always meet at a right angle, so we always know the angle involved between them.


Fun Facts

  • Congruent right triangles appear like a marching band or tuba players just how they have the same uniforms, and similar organized patterns of marching.

  • With right triangles, you always obtain a "freebie" identifiable angle, in every congruence

FAQ (Frequently Asked Questions)

1. What Is Meant By Right Angle Triangle Congruence Theorem?

With Right triangles, it is meant that one of the interior angles in a triangle will be 90 degrees, which is called a right angle. Considering that the sum of all the 3 interior angles of a triangle add up to 180°, in a right triangle, and that only one angle is always 90°, the other two should always add up to 90° (they are supplementary).       

That said, All right triangles are with two legs, which may or may not be similar in size. The legs of a right triangle touch at a right angle. The other side of the triangle (that does not develop any portion of the right angle), is known as the hypotenuse of the right triangle. This side of the right triangle (hypotenuse) is unquestionably the longest of all three sides always. Keep in mind that the angles of a right triangle that are not the right angle should be acute angles.