Triangle Congruence - SSS and SAS
We have learned that triangles are congruent if their corresponding sides and angles are congruent. However, there are excessive requirements that need to be met in order for this claim to hold. In this section, we will learn two postulates that prove triangles congruent with less information required. These postulates are useful because they only require three corresponding parts of triangles to be congruent (rather than six corresponding parts like with CPCTC). Let's take a look at the first postulate.
SSS Postulate (Side-Side-Side)
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
As you can see, the SSS Postulate does not concern itself with angles at all. Rather, it only focuses only on corresponding, congruent sides of triangles in order to determine that two triangles are congruent. An illustration of this postulate is shown below.
We conclude that ?ABC??DEF because all three corresponding sides of the triangles are congruent.
Let's work through an exercise that requires the use of the SSS Postulate.
The only information that we are given that requires no extensive work is that segment JK is congruent to segment NK. We are given the fact that A is a midpoint, but we will have to analyze this information to derive facts that will be useful to us.
In the two triangles shown above, we only have one pair of corresponding sides that are equal. However, we can say that AK is equal to itself by the Reflexive Property to give two more corresponding sides of the triangles that are congruent.
Finally, we must make something of the fact A is the midpoint of JN. By definition, the midpoint of a line segment lies in the exact middle of a segment, so we can conclude that JA?NA.
After doing some work on our original diagram, we should have a figure that looks like this:
Now, we have three sides of a triangle that are congruent to three sides of another triangle, so by the SSS Postulate, we conclude that ?JAK??NAK. Our two column proof is shown below.
We involved no angles in the SSS Postulate, but there are postulates that do include angles. Let's take a look at one of these postulates now.
SAS Postulate (Side-Angle-Side)
If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
A key component of this postulate (that is easy to get mistaken) is that the angle must be formed by the two pairs of congruent, corresponding sides of the triangles. If the angles are not formed by the two sides that are congruent and corresponding to the other triangle's parts, then we cannot use the SAS Postulate. We show a correct and incorrect use of this postulate below.
The diagram above uses the SAS Postulate incorrectly because the angles that are congruent are not formed by the congruent sides of the triangle.
The diagram above uses the SAS Postulate correctly. Notice that the angles that are congruent are formed by the corresponding sides of the triangle that are congruent.
Let's use the SAS Postulate to prove our claim in this next exercise.
For this solution, we will try to prove that the triangles are congruent by the SAS Postulate. We are initially given that segments AC and EC are congruent, and that segment BC is congruent to DC.
If we can find a way to prove that ?ACB and ?ECD are congruent, we will be able to prove that the triangles are congruent because we will have two corresponding sides that are congruent, as well as congruent included angles. Trying to prove congruence between any other angles would not allow us to apply the SAS Postulate.
The way in which we can prove that ?ACB and ?ECD are congruent is by applying the Vertical Angles Theorem. This theorem states that vertical angles are congruent, so we know that ?ACB and ?ECD have the same measure. Our figure show look like this:
Now we have two pairs of corresponding, congruent sides, as well as congruent included angles. Applying the SAS Postulate proves that ?ABC??EDC. The two-column geometric proof for our argument is shown below.
Angle Properties of Triangles
Now that we are acquainted with the classifications of triangles, we can begin our extensive study of the angles of triangles. In many cases, we will have to utilize the angle theorems we've seen to help us solve problems and proofs. However, there are some triangle theorems that will be just as essential to know. This first theorem tells us that if we know the measures of two angles of a triangle, it is possible to determine the measure of the third angle.
Triangle Angle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180.
The diagram above illustrates the Triangle Angle Sum Theorem.
Let's do some examples involving the Triangle Sum Theorem to help us see its utility.
(1) Find the measure of ?C.
As with all problems, we must first use the facts that are given to us. Using the diagram, we are given that
Since our goal is to find the measure of ?C, we can use the Triangle Angle Sum Theorem to solve for the missing angle. So we have
Using the angle measures we were given, we can substitute those values into our equation to get.
Having ?C measure out to 26° satisfies the property that the sum of the interior angles of a triangle is 180°.
(2) Find the value of x in the diagram below.
In this exercise, we are given that
Looking at ?RST, we see that two of three angles are given to us. Thus, we can apply the Triangle Angle Sum Theorem to figure out the measure of the third angle:
Note that ?SRT is the vertical angle opposite ?QRP, so we can deduce that
Then, by the definition of congruent angles, we have
Now, we have one of three angle measures of ?QRP. Since we know that m?P = m?Q = x, we can use the Triangle Angle Sum Theorem as follows
We have found the measure of ?P and ?Q to be 67.
In order to comprehend the next theorem, we must learn two more terms that describe angles. The angle formed by one side of a triangle with the extension of another side is called an exterior angle of the triangle.
Exterior angles get their name because they lie on the outsides of triangles.
The two angles that are not adjacent, or next to, the exterior angle of the triangle are called remote interior angles.
Now that we know what these terms mean, we are ready for a theorem that will help us tremendously in our proofs.
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Adding the measures of the two remote interior angles of a triangle gives the measure of the exterior angle.
Let's see how the Exterior Angle Theorem can be utilized to help us find the measures of unknown angles in the examples below.
(1) Find the measures of ?1 and ?2 in the figure below.
First, we can solve for m?1 since we are given the measure of two angles of that triangle. This part of the problem is similar to the examples we have already done above. Let's begin with the statements of what we are given, which are:
Now, we can solve for m?1 by using the Triangle Angle Sum Theorem. So we have
In order to solve for the measure of ?2, we will need to apply the Exterior Angle Theorem. We know that the two remote interior angles in the figure are ?S and ?A. Thus, by the Exterior Angle Theorem, the sum of those angles is equal to the measure of the exterior angle. We have
While not always necessary, we can check our solution using our previous knowledge of lines. We see that ?1 and ?2 make up ray AK. And since straight lines have 180° measures, we know that the sum of ?1 and ?2 must be 180. Let's check to make sure:
So, we know we have worked this problem out correctly.
(2) Find m?B.
Let's take a look at the information we have been given first. We know that
Right off the bat, we can apply the Exterior Angle Theorem to help us solve the problem. We have
This does not answer the question, however. The question asked for m?B. The variable x alone does not tell us what the measure of the angle is. So, we must plug x = 4 into our equation for m?B:
Now we have found that the measure of ?B is 39°.