Right Triangle Congruence
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?DE) and congruent legs (CA?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.
What additional information do we need in order to prove that the triangles below are congruent by the Hypotenuse-Leg Theorem?
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 ?QRS??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.
In which of the following figures could we use the Hypotenuse-Leg Theorem to show that the triangles are congruent?
Let's take a closer look at all of the diagrams to determine which of them show a pair of congruent triangles by the HL Theorem.
In (a), it appears as though we might be able to use the HL Theorem. However, upon careful examination, we note that the angles at vertices A and D are not right angles. Because a square is not used to indicate that the angles are right angles, we cannot use the HL Theorem. Recall that the only type of triangle for which this theorem holds is a right triangle, so we cannot apply it in this situation.
Figure (b) does show two triangles that are congruent, but not by the HL Theorem. We are given that segment FG is congruent to segment HG and that segment EG is congruent to segment IG. We also have right angles that form at G. Because we have two sides and the included angle of one triangle congruent to the corresponding parts of the other triangle, we know that the triangles are congruent by the SAS Postulate. However, we are not given any information regarding the hypotenuses of ?EGF and ?IHG, so we cannot apply the HL Theorem to prove that the triangles are congruent.
Now, let's look at (c). Notice that we have two right angles in the figure: ?JLK and ?JLM. Also, we have been given the fact that segment JK is congruent to segment JM. These segments are actually the hypotenuses of the triangles because they lie on the side opposite of the right angle. Moreover, the two triangles in the figure share segment JL. By transitivity, we know that the segment is congruent to itself. Thus, we can apply the HL Theorem to prove that ?JKL??JML, since we know that the triangles are right triangles, their hypotenuses are congruent, and they have a pair of legs that are congruent.
Finally, we have the figure for (d). We have been given that there are right angles at vertices O and Q. We can also imply that ?NPO and ?RPQ are congruent because they are vertical angles. This will not help us try to prove that the triangles are congruent by the HL Theorem, however. What we are looking for is information about the legs or hypotenuses of the triangles. Since we cannot deduce any more facts from the diagram that will help us, we cannot apply the HL Theorem in this situation.
Therefore, we can only apply the HL Theorem in (c) to show that the triangles are congruent.
We want to examine the information that has been given to us in the problem. We know that segment RV is perpendicular to segment SK, and that segments SR and KR are congruent. Let's try to deduce more information from the given statements that may help us prove that ?RSV??RKV.
Since we were given that RV and SK are perpendicular, we know that there exist right angles at ?RVS and ?RVK. This fact is a key component of our proof because we know that ?RSV and ?RKV are right triangles. Thus, we can try to use the HL Theorem to prove that they are congruent to each other.
We have already been given that the hypotenuses are congruent, so all that is left to show is that a pair of legs of the triangles is congruent. Since they both share segment RV, we can use the Transitive Property to say that the segment is congruent to itself.
In all, we have found right angles, congruent hypotenuses, and congruent legs between the triangles, so we apply the HL Theorem to say that ?RSV??RKV. Our new 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, which we will not study in detail because they are equivalent to the congruence postulates we've already learned. 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.