A very important topic in the study of geometry is congruence. Thus far, we have only learned about congruent angles, but in this section we will learn about the criteria necessary for triangles to be congruent. Learning about congruence on this level will open the door to different triangle congruence theorems that characterize geometry.
Recall that in order for lines or angles to be congruent, they had to have equal measures. In that same way, congruent triangles are triangles with corresponding sides and angles that are congruent, giving them the same size and shape. Because side and angle correspondence is important, we have to be careful with the way we name triangles. For instance, if we have ?ABC??DEF, the congruence between triangles implies the following:
It is important to name triangles correctly to identify which segments are equal in length and to see which angles have the same measures.
In short, we say that two triangles are congruent if their corresponding parts (which include lines and angles) are congruent. In a two-column geometric proof, we could explain congruence between triangles by saying that "corresponding parts of congruent triangles are congruent." This statement is rather long, however, so we can just write "CPCTC" for short.
Third Angles Theorem
In some instances we will need a very significant theorem to help us prove congruence between two triangles. If we know that two angles of two separate triangles are congruent, our inclination is to believe that their third angles are equal because of the Triangle Angle Sum Theorem.
This type of reasoning is correct and is a very helpful theorem to use when trying to prove congruence between triangles. The Third Angles Theorem states that if two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent also.
Let's take a look at some exercises to put our knowledge of congruent triangles, CPCTC, and the Third Angles Theorem to work.
(1) Which of the following expresses the correct congruence statement for the figure below?
While it may not seem important, the order in which you list the vertices of a triangle is very significant when trying to establish congruence between two triangles. Essentially what we want to do is find the answer that helps us correspond the triangles' points, sides, and angles. The answer that corresponds these characteristics of the triangles is (b).
In answer (b), we see that ?PQR ? ?LJK. Let's start off by comparing the vertices of the triangles. In the first triangle, the point P is listed first. This corresponds to the point L on the other triangle. We know that these points match up because congruent angles are shown at those points. Listed next in the first triangle is point Q. We compare this to point J of the second triangle. Again, these match up because the angles at those points are congruent. Finally, we look at the points R and K. The angles at those points are congruent as well.
We can also look at the sides of the triangles to see if they correspond. For instance, we could compare side PQ to side LJ. The figure indicates that those sides of the triangles are congruent. We can also look at two more pairs of sides to make sure that they correspond. Sides QR and JK have three tick marks each, which shows that they are congruent. Finally, sides RP and KJ are congruent in the figure. Thus, the correct congruence statement is shown in (b).
(2) Find the values of x and y given that ?MAS ? ?NER.
We have two variables we need to solve for. It would be easiest to use the 16x to solve for x first (because it is a single-variable expression), as opposed to using the side NR, would require us to try to solve for x and y at the same time. We must look for the angle that correspond to ?E so we can set the measures equal to each other. The angle that corresponds to ?E is ?A, so we get
Now that we have solved for x, we must use it to help us solve for y. The side that RN corresponds to is SM, so we go through a similar process like we did before.
Now we substitute 7 for x to solve for y:
We have finished solving for the desired variables.
To begin this problem, we must be conscious of the information that has been given to us. We know that two pairs of sides are congruent and that one set of angles is congruent. In order to prove the congruence of ?RQT and ?SQT, we must show that the three pairs of sides and the three pairs of angles are congruent.
Since QS is shared by both triangles, we can use the Reflexive Property to show that the segment is congruent to itself. We have now proven congruence between the three pairs of sides. The congruence of the other two pairs of sides were already given to us, so we are done proving congruence between the sides.
Now we must show that all angles are congruent within the triangles. One pair has already been given to us, so we must show that the other two pairs are congruent. It has been given to us that QT bisects ?RQS. By the definition of an angle bisector, we know that two equivalent angles exist at vertex Q. The final pairs of angles are congruent by the Third Angles Theorem (since the other two pairs of corresponding angles of the triangles were congruent). We conclude that the triangles are congruent because corresponding parts of congruent triangles are congruent. The two-column geometric proof that shows our reasoning is below.
We are given that the three pairs of corresponding sides are congruent, so we do not have to worry about this part of the problem; we only need to worry about proving congruence between corresponding angles.
We are only given that one pair of corresponding angles is congruent, so we must determine a way to prove that the other two pairs of corresponding angles are congruent. We do this by showing that ?ACB and ?ECD are vertical angles. So, by the Vertical Angles Theorem, we know that they are congruent to each other. Now that we know that two of the three pairs of corresponding angles of the triangles are congruent, we can use the Third Angles Theorem. This theorem states that if we have two pairs of corresponding angles that are congruent, then the third pair must also be congruent.
Since all three pairs of sides and angles have been proven to be congruent, we know the two triangles are congruent by CPCTC. The two-column geometric proof that shows our reasoning is below.
Triangle Congruence - ASA and AAS
We've just studied two postulates that will help us prove congruence between triangles. However, these postulates were quite reliant on the use of congruent sides. In this section, we will get introduced to two postulates that involve the angles of triangles much more than the SSS Postulate and the SAS Postulate did. Understanding these four postulates and being able to apply them in the correct situations will help us tremendously as we continue our study of geometry. Let's take a look at our next postulate.
ASA Postulate (Angle-Side-Angle)
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
In a sense, this is basically the opposite of the SAS Postulate. The SAS Postulate required congruence of two sides and the included angle, whereas the ASA Postulate requires two angles and the included side to be congruent. An illustration of this postulate is shown below.
We conclude that ?ABC??DEF by the ASA Postulate because the triangles' two angles and included side are congruent.
Let's practice using the ASA Postulate to prove congruence between two triangles.
Let's start off this problem by examining the information we have been given. Since segments PQ and RS are parallel, this tells us that we may need to use some of the angle postulates we've studied in the past. Now, let's look at the other piece of information we've been given. We know that ?PRQ is congruent to ?SQR. Let's further develop our plan of attack.
We have been given just one pair of congruent angles, so let's look for another pair that we can prove to be congruent. We can say ?PQR is congruent to ?SQR by the Alternate Interior Angles Postulate. Recall, we can only use this postulate when a transversal crosses a set of parallel lines. In this case, our transversal is segment RQ and our parallel lines have been given to us.
Now that we've established congruence between two pairs of angles, let's try to do something with the included side. The included side is segment RQ. By using the Reflexive Property to show that the segment is equal to itself, we now have two pairs of congruent angles, and common shared line between the angles. Our new illustration is shown below.
We conclude our proof by using the ASA Postulate to show that ?PQR??SRQ. Let's look at our two-column geometric proof that shows the arguments we've made.
Aside from the ASA Postulate, there is also another congruence postulate that involves two pairs of congruent angles and one pair of congruent sides. Let's take a look at this postulate now.
AAS Postulate (Angle-Angle-Side)
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
In order to use this postulate, it is essential that the congruent sides not be included between the two pairs of congruent angles. If the side is included between the angles, we would actually need to use the ASA Postulate. The correct use of the AAS Postulate is shown below.
We conclude that ?ABC??DEF by the AAS Postulate since we have two pairs of congruent angles and one pair of congruent sides not included between the angles.
Let's use the AAS Postulate to prove the claim in our next exercise.
Before we begin our proof, let's see how the given information can help us. We have been given that ?NER??NVR, so that is one pair of angles that we do not need to show as congruent.
Now, we must decide on which other angles to show congruence for. We may be able to derive a key component of this proof from the second piece of information given. Since segment RN bisects ?ERV, we can show that two congruent angles are formed. By the definition of an angle bisector, we have that ?ERN??VRN.
The only component of the proof we have left to show is that the triangles have congruent sides. Luckily for us, the triangles are attached by segment RN. So, we use the Reflexive Property to show that RN is equal to itself. Let's look at our new figure.
Finally, by the AAS Postulate, we can say that ?ENR??VNR. Note that our side RN is not included. If it were included, we would use the ASA Postulate to prove that the triangles are congruent. The two-column proof for this exercise is shown below.