Determine if the two triangles are congruent. if they are, state the postulate.

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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.

Determine if the two triangles are congruent. if they are, state the postulate.

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)

Determine if the two triangles are congruent. if they are, state the postulate.

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.

Determine if the two triangles are congruent. if they are, state the postulate.

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.

Determine if the two triangles are congruent. if they are, state the postulate.

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.

Determine if the two triangles are congruent. if they are, state the postulate.

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?

Determine if the two triangles are congruent. if they are, state the postulate.

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

Congruent triangles are triangles with identical sides and angles. The three sides of one are exactly equal in measure to the three sides of another. The three angles of one are each the same angle as the other.

Triangle Congruence Postulates

Five ways are available for finding two triangles congruent:

  1. SSS, or Side Side Side
  2. SAS, or Side Angle Side
  3. ASA, or Angle Side Side
  4. AAS, or Angle Angle Side
  5. HL, or Hypotenuse Leg, for right triangles only

Included Parts

An included angle lies between two named sides. In △CAT below, included ∠A is between sides t and c:

Determine if the two triangles are congruent. if they are, state the postulate.

An included side lies between two named angles of the triangle.

Side Side Side Postulate

A postulate is a statement taken to be true without proof. The SSS Postulate tells us,

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

Congruence of sides is shown with little hatch marks, like this: ∥. For two triangles, sides may be marked with one, two, and three hatch marks.

If △ACE has sides identical in measure to the three sides of △HUM, then the two triangles are congruent by SSS:

Determine if the two triangles are congruent. if they are, state the postulate.

Side Angle Side Postulate

The SAS Postulate tells us,

If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

△HUG and △LAB each have one angle measuring exactly 63°. Corresponding sides g and b are congruent. Sides h and l are congruent.

A side, an included angle, and a side on △HUG and on △LAB are congruent. So, by SAS, the two triangles are congruent.

Determine if the two triangles are congruent. if they are, state the postulate.

Angle Side Angle Postulate

This postulate says,

If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

We have △MAC and △CHZ, with side m congruent to side c. ∠A is congruent to ∠H, while ∠C is congruent to ∠Z. By the ASA Postulate these two triangles are congruent.

Determine if the two triangles are congruent. if they are, state the postulate.

Angle Angle Side Theorem

We are given two angles and the non-included side, the side opposite one of the angles. The Angle Angle Side Theorem says,

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

Here are congruent △POT and △LID, with two measured angles of 56° and 52°, and a non-included side of 13 centimeters:

[construct as described]

By the AAS Theorem, these two triangles are congruent.

HL Postulate

Exclusively for right triangles, the HL Postulate tells us,

Two right triangles that have a congruent hypotenuse and a corresponding congruent leg are congruent.

The hypotenuse of a right triangle is the longest side. The other two sides are legs. Either leg can be congruent between the two triangles.

Here are right triangles △COW and △PIG, with hypotenuses of sides w and i congruent. Legs o and g are also congruent:

[insert congruent right triangles left-facing △COW and right facing △PIG]

So, by the HL Postulate, these two triangles are congruent, even if they are facing in different directions.

Proof Using Congruence

Determine if the two triangles are congruent. if they are, state the postulate.

Given: △MAG and △ICG

MC ≅ AI

AG ≅ GI

Prove: △MAG ≅ △ICG

Statement Reason

MC ≅ AI Given

AG ≅ GI

∠MGA ≅ ∠ IGC Vertical Angles are Congruent

△MAG ≅ △ICG Side Angle Side

If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Next Lesson:

Triangle Congruence Theorems

Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.