What additional information is needed to prove that the triangles are congruent using the SAS congruence theorem?

Triangles can be similar or congruent. Similar triangles will have congruent angles but sides of different lengths. Congruent triangles will have completely matching angles and sides. Their interior angles and sides will be congruent. Testing to see if triangles are congruent involves three postulates, abbreviated SAS, ASA, and SSS.

Congruence Definition

Two triangles are congruent if their corresponding sides are equal in length and their corresponding interior angles are equal in measure.

We use the symbol ≅ to show congruence.

Corresponding sides and angles mean that the side on one triangle and the side on the other triangle, in the same position, match. You may have to rotate one triangle, to make a careful comparison and find corresponding parts.

How can you tell if triangles are congruent?

You could cut up your textbook with scissors to check two triangles. That is not very helpful, and it ruins your textbook. If you are working with an online textbook, you cannot even do that.

Geometricians prefer more elegant ways to prove congruence. Comparing one triangle with another for congruence, they use three postulates.

Postulate Definition

A postulate is a statement presented mathematically that is assumed to be true. All three triangle congruence statements are generally regarded in the mathematics world as postulates, but some authorities identify them as theorems (able to be proved).

Do not worry if some texts call them postulates and some mathematicians call the theorems. More important than those two words are the concepts about congruence.

Triangle Congruence Theorems

Testing to see if triangles are congruent involves three postulates. Let's take a look at the three postulates abbreviated ASA, SAS, and SSS.

1. Angle Side Angle (ASA)
2. Side Angle Side (SAS)
3. Side Side Side (SSS)

ASA Theorem (Angle-Side-Angle)

The Angle Side Angle Postulate (ASA) says triangles are congruent if any two angles and their included side are equal in the triangles. An included side is the side between two angles.

In the sketch below, we have △CAT and △BUG. Notice that ∠C on △CAT is congruent to ∠B on △BUG, and ∠A on △CAT is congruent to ∠U on △BUG.

See the included side between ∠C and ∠A on △CAT? It is equal in length to the included side between ∠B and ∠U on △BUG.

The two triangles have two angles congruent (equal) and the included side between those angles congruent. This forces the remaining angle on our △CAT to be:

180° - ∠C - ∠A

This is because interior angles of triangles add to 180°. You can only make one triangle (or its reflection) with given sides and angles.

You may think we rigged this, because we forced you to look at particular angles. The postulate says you can pick any two angles and their included side. So go ahead; look at either ∠C and ∠T or ∠A and ∠T on △CAT.

Compare them to the corresponding angles on △BUG. You will see that all the angles and all the sides are congruent in the two triangles, no matter which ones you pick to compare.

SAS Theorem (Side-Angle-Side)

By applying the Side Angle Side Postulate (SAS), you can also be sure your two triangles are congruent. Here, instead of picking two angles, we pick a side and its corresponding side on two triangles.

The SAS Postulate says that triangles are congruent if any pair of corresponding sides and their included angle are congruent.

Pick any side of △JOB below. Notice we are not forcing you to pick a particular side, because we know this works no matter where you start. Move to the next side (in whichever direction you want to move), which will sweep up an included angle.

For the two triangles to be congruent, those three parts -- a side, included angle, and adjacent side -- must be congruent to the same three parts -- the corresponding side, angle and side -- on the other triangle, △YAK.

SSS Theorem (Side-Side-Side)

Perhaps the easiest of the three postulates, Side Side Side Postulate (SSS) says triangles are congruent if three sides of one triangle are congruent to the corresponding sides of the other triangle.

This is the only postulate that does not deal with angles. You can replicate the SSS Postulate using two straight objects -- uncooked spaghetti or plastic stirrers work great. Cut a tiny bit off one, so it is not quite as long as it started out. Cut the other length into two distinctly unequal parts. Now you have three sides of a triangle. Put them together. You have one triangle. Now shuffle the sides around and try to put them together in a different way, to make a different triangle.

Guess what? You can't do it. You can only assemble your triangle in one way, no matter what you do. You can think you are clever and switch two sides around, but then all you have is a reflection (a mirror image) of the original.

So once you realize that three lengths can only make one triangle, you can see that two triangles with their three sides corresponding to each other are identical, or congruent.

You can check polygons like parallelograms, squares and rectangles using these postulates.

Introducing a diagonal into any of those shapes creates two triangles. Using any postulate, you will find that the two created triangles are always congruent.

Suppose you have parallelogram SWAN and add diagonal SA. You now have two triangles, △SAN and △SWA. Are they congruent?

You already know line SA, used in both triangles, is congruent to itself. What about ∠SAN? It is congruent to ∠WSA because they are alternate interior angles of the parallel line segments SW and NA (because of the Alternate Interior Angles Theorem).

You also know that line segments SW and NA are congruent, because they were part of the parallelogram (opposite sides are parallel and congruent).

So now you have a side SA, an included angle ∠WSA, and a side SW of △SWA. You can compare those three triangle parts to the corresponding parts of △SAN:

• Side SA ≅ Side SA   (sure hope so!)
• Included angle   ∠WSA ≅ ∠NAS
• Side SW ≅ Side NA

Lesson Summary

After working your way through this lesson and giving it some thought, you now are able to recall and apply three triangle congruence postulates, the Side Angle Side Congruence Postulate, Angle Side Angle Congruence Postulate, and the Side Side Side Congruence Postulate. You can now determine if any two triangles are congruent!

Next Lesson:

Conditional Statements and Their Converse

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS   (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS   (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:

is congruent to:

(See Solving SAS Triangles to find out more)

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA   (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:

is congruent to:

(See Solving ASA Triangles to find out more)

If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS   (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

is congruent to:

(See Solving AAS Triangles to find out more)

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL   (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for "Hypotenuse, Leg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

• the same length of hypotenuse and
• the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:

is congruent to:

(See Pythagoras' Theorem to find out more)

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.

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