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.