Geometry (A): READ: Triangle Congruence/ SSS SAS Reading

Triangle Congruence using SSS and SAS

Learning Objectives

Use the distance formula to analyze triangles on a coordinate grid.
Understand and apply the SSS postulate of triangle congruence.

Introduction

In the last section you learned that if two triangles are congruent then the three pairs of corresponding sides are congruent and the three pairs of corresponding angles are congruent. In symbols, $\triangle{CAT} \cong \triangle{DOG}$ means $\angle{C} \cong \angle{D}, \angle{A} \cong \angle{O}, \angle{T} \cong \angle{G}, \overline{GA} \cong \overline{DO}, \overline{AT} \cong \overline{OG}$ , and $\overline{CT} \cong \overline{DG}$ .

Wow, that’s a lot of information—in fact, one triangle congruence statement contains six different congruence statements! In this section we show that proving two triangles are congruent does not necessarily require showing all six congruence statements are true. Lucky for us, there are shortcuts for showing two triangles are congruent—this section and the next explore some of these shortcuts.

SSS Postulate of Triangle Congruence

The SSS postulate states that when three sides of one triangle are equal in length to three sides of another, then the triangles are congruent. We did not need to measure the angles—the lengths of the corresponding sides being the same “forced” the corresponding angles to be congruent. This leads us to one of the triangle congruence postulates:

Side-Side-Side (SSS) Triangle Congruence Postulate: If three sides in one triangle are congruent to the three corresponding sides in another triangle, then the triangles are congruent to each other.

This is a postulate so we accept it as true without proof.

You can perform a quick experiment to test this postulate. Cut two pieces of spaghetti (or a straw, or some segment-like thing) exactly the same length. Then cut another set of pieces that are the same length as each other (but not necessarily the same length as the first set). Finally, cut one more pair of pieces of spaghetti that are identical to each other. Separate the pieces into two piles. Each pile should have three pieces of different lengths. Build a triangle with one set and leave it on your desk. Using the other pieces, attempt to make a triangle with a different shape or size by matching the ends. Notice that no matter what you do, you will always end up with congruent triangles (though they might be “flipped over” or rotated). This demonstrates that if you can identify three pairs of congruent sides in two triangles, the two triangles are fully congruent.

Example 2

Write a triangle congruence statement based on the diagram below:

We can see from the tick marks that there are three pairs of corresponding congruent sides: $\overline{HA} \cong \overline{RS}$ , $\overline{AT} \cong \overline{SI}$ , and $\overline{TH} \cong \overline{IR}$ . Matching up the corresponding sides, we can write the congruence statement $\triangle{HAT} \cong \triangle{RSI}$ .

Don’t forget that ORDER MATTERS when writing triangle congruence statements. Here, we lined up the sides with one tic mark, then the sides with two tic marks, and finally the sides with three tic marks.

SAS Congruence

By now, you are very familiar with postulates and theorems using the letters $S$ and $A$ to represent triangle sides and angles. One more way to show two triangles are congruent is by the SAS Congruence Postulate.

SAS Triangle Congruence Postulate: If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent.

Like ASA and AAS congruence, the order of the letters is very significant. You must have the angles between the two sides for the SAS postulate to be valid.

Once again you can test this postulate using physical models (such as pieces of uncooked spaghetti) for the sides of a triangle. You’ll find that if you make two pairs of congruent sides, and lay them out with the same included angle then the third side will be determined.

Example 1

What information would you need to prove that these two triangles were congruent using the SAS postulate?

A. the measures of $\angle{HJG}$ and $\angle{STR}$

B. the measures of $\angle{HGJ}$ and $\angle{SRT}$

C. the measures of $\overline{HJ}$ and $\overline{ST}$

D. the measures of sides $\overline{GJ}$ and $\overline{RT}$

If you are to use the SAS postulate to establish congruence, you need to have the measures of two sides and the angle in between them for both triangles. So far, you have one side and one angle. So, you must use the other side adjacent to the same angle. In $\triangle{GHJ}$ , that side is $\overline{HJ}$ . In triangle $\triangle{RST}$ , the corresponding side is $\overline{ST}$ . So, the correct answer is C.

AAA and SSA relationships

You have learned so many different ways to prove congruence between two triangles, it may be tempting to think that if you have any pairs of congruent three elements (combining sides or angles), you can prove triangle congruence.

However, you may have already guessed that AAA congruence does not work. Even if all of the angles are equal between two triangles, the triangles may be of different scales. So, AAA can only prove similarity, not congruence.

SSA relationships do not necessarily prove congruence either. Get your spaghetti and protractors back on your desk to try the following experiment. Choose two pieces of spaghetti at given length. Select a measure for an angle that is not between the two sides. If you keep that angle constant, you may be able to make two different triangles. As the angle in between the two given sides grows, so does the side opposite it. In other words, if you have two sides and an angle that is not between them, you cannot prove congruence.

In the figure, $\triangle{ABC}$ is NOT congruent to $\triangle{FEH}$ even though they have two pairs of congruent sides and a pair of congruent angles. $\overline{FG} \cong \overline{FH} \cong \overline{AC}$ and you can see that there are two possible triangles that can be made using this combination SSA.

Example 2

Can you prove that the two triangles below are congruent?

Note: Figure is not to scale.

The two triangles above look congruent, but are labeled, so you cannot assume that how they look means that they are congruent. There are two sides labeled congruent, as well as one angle. Since the angle is not between the two sides, however, this is a case of SSA. You cannot prove that these two triangles are congruent. Also, it is important to note that although two of the angles appear to be right angles, they are not marked that way, so you cannot assume that they are right angles.

Lesson Summary

In this lesson, we explored triangle congruence using only the sides. Specifically, we have learned:

How to use the distance formula to analyze triangles on a coordinate grid
How to understand and apply the SSS postulate of triangle congruence.

These skills will help you understand issues of congruence involving triangles, and later you will apply this knowledge to all types of shapes.

Points to Consider

Now that you have been exposed to the SSS Postulate, there are other triangle congruence postulates to explore. The next chapter deals with congruence using a mixture of sides and angles.

The following questions are for your own benefit. The answers are below for you to check your work and understanding.

Review Questions

If you know that $\triangle PQR \cong \triangle STU$ in the diagram below, what are six congruence statements that you also know about the parts of these triangles?
Redraw these triangles using geometric markings to show all congruent parts.

Use the diagram below for exercises 3-7 .

Find the length of each side in
1. $AB =$
2. $BC =$
3. $AC =$
Find the length of each side in
1. $XY =$
2. $YZ =$
3. $XZ =$
Write a congruence statement relating these two triangles.
Write another equivalent congruence statement for these two triangles.
What postulate guarantees these triangles are congruent?

Exercises 8-10 use the following diagram:

Write a congruence statement for the two triangles in this diagram. What postulate did you use?
Find $m \angle{C}$ . Explain how you know your answer.
Find $m \angle{R}$ . Explain how you know your answer.

Review Answers

$\overline{PQ} \cong \overline{ST}, \overline{QR} \cong \overline{TU}, \overline{PR} \cong \overline{SU}, \angle{P} \cong \angle {S}, \angle{Q} \cong \angle{T},$ and $\angle{R} \cong \angle{U}$
One possible answer:
1. $AB = 5 \;\mathrm{units}$
2. $BC = 3\sqrt{2} \;\mathrm{units}$
3. $AC = \sqrt{37} \;\mathrm{units}$
1. $XY = 3\sqrt{2} \;\mathrm{units}$
2. $YZ = 5 \;\mathrm{units}$
3. $XZ = \sqrt{37} \;\mathrm{units}$
$\triangle ABC \cong \triangle ZYX$ (Note, other answers are possible, but the relative order of the letters does matter.)
$\triangle BAC \cong \triangle YZX$
SSS
$\triangle ABC \cong \triangle RIT$ , the side-side-side triangle congruence postulate
$m\angle{c} = 39^\circ$ . We know this because it corresponds with $\angle{T}$ , so $\angle{C} \cong \angle{T}$
$m\angle{R} = 75^\circ$ . Used the triangle sum theorem together with my answer for 9.

Last modified: Monday, June 28, 2010, 2:23 PM