Download File

Right Triangle Congruence: HL, HA, LL, and LA
Isosceles and equilateral triangles aren't the only classifications of triangles with special characteristics. Right triangles are also
significant in the study of geometry and, as we will see, we will be able to prove the congruence of right triangles in an efficient
way.
Before we begin learning this, however, it is important to break down right triangles into parts. Learning terms that refer to the
parts of a right triangle will help us avoid confusion throughout this section.
All right triangles have two legs, which may or may not be congruent. The legs of a right triangle meet at a right angle. The
other side of the triangle (that does not form any part of the right angle), is called the hypotenuse of the right triangle. This side
of the right triangle will always be the longest of all three sides. The angles of a right triangle that are not the right angle must be
acute angles.
Now, let's learn what the Hypotenuse-Leg Theorem is and how to apply it.
Hypotenuse-Leg (HL) Theorem
If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two
right triangles are congruent.
Recall that the criteria for our congruence postulates have called for three pairs of congruent parts between triangles. The HL
Theorem essentially just calls for congruence between two parts: the hypotenuse and a leg. Let's look at an illustration that shows
the correct way to use the Hypotenuse-Leg Theorem.
In the figure, we have congruent hypotenuses (AB and DE) and congruent legs (CA and FD).
We are ready to begin practicing with the HL Theorem. Let's go through the following exercises to get a feel for how to use this
helpful theorem.
Exercise 1
What additional information do we need in order to prove that the triangles below are congruent by the Hypotenuse-Leg
Theorem?
Answer:
Notice that both triangles are right triangles because they both have one right angle in them. Therefore, if we can prove that the
hypotenuses of the triangles and one leg of each triangle are congruent, we will be able to apply the HL Theorem.
Looking at the diagram, we notice that segments SQ and VT are congruent. Recall that the side of a right triangle that does not
form any part of the right angle is called the hypotenuse. So, the diagram shows that we have congruent hypotenuses.
No other information about the triangles is given to us, though. Had we been given that another pair of legs was congruent, then
our criteria for using the HL Theorem would have been satisfied. Below, we show two situations in which we could have used
the HL Theorem to prove that Triangle QRS is congruent to Triangle TUV.
In the diagram above, we are given all of the same information as in the original, as well as the fact that segments QR and TU
are congruent. We could have applied the HL Theorem in this situation to prove congruence.
In the diagram above, we note that all of the original information has been given to us as well as the fact that RS and UV are
congruent. Here, we could have applied the HL Theorem to prove that the triangles are congruent.
Exercise 2
In which of the following figures could we use the Hypotenuse-Leg Theorem to show that the triangles are congruent?
(a)
(b)
(c)
(d)
Answer:
We can only apply the HL Theorem in (c) to show that the triangles are congruent.
Exercise 3
Answer: diagram and the two-column geometric proof are shown below.
The HL Theorem will be used throughout the rest of our study of geometry. There are other theorems that are specific to right
triangles. These theorems and their equivalent postulates are explained below.
Leg-Leg (LL) Theorem
If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent.
This statement is the same as the SAS Postulate we've learned about because it involves two sides of triangles, as well as the
included angle (which is the right angle).
Leg-Acute (LA) Angle Theorem
If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the two
right triangles are congruent.
This statement is equivalent to the ASA Postulate we've learned about because it involves right angles (which are congruent), a
pair of sides with the same measure, and congruent acute angles.
Hypotenuse-Acute (HA) Angle Theorem
If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right
triangle, then the two triangles are congruent.
This statement is the same as the AAS Postulate because it includes right angles (which are congruent), two congruent acute
angles, and a pair of congruent hypotenuses.