Triangle Congruence Theorems (SSS, SAS, & ASA Postulates)

Triangles deserve to be comparable or congruent. Similar triangles will have congruent angles yet sides of different lengths. Congruent triangles will have completely matching angles and also sides. Their inner angles and sides will certainly be congruent. Experimentation to watch if triangles room congruent requires three postulates, abbreviated SAS, ASA, and SSS.

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Congruence Definition

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


We usage the symbol ≅ to display congruence.

Corresponding sides and angles median that the next on one triangle and also the side on the other triangle, in the exact same position, match. You may have to rotate one triangle, to do a cautious comparison and find matching parts.

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How can you phone call if triangles space congruent?

You could cut up her textbook v scissors to inspect two triangles. The is not very helpful, and also it ruins your textbook. If you room working v an virtual textbook, you cannot even do that.

Geometricians prefer more elegant means to prove congruence. To compare one triangle with another for congruence, castle use three postulates.

Postulate Definition

A postulate is a statement gift mathematically the is suspect to it is in true. All three triangle congruence statements are usually regarded in the mathematics people as postulates, however some authorities recognize them as theorems (able to be proved).


Do not problem if part texts call them postulates and some mathematicians call the theorems. Much more important than those two words room 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 abbreviation ASA, SAS, and SSS.

Angle next Angle (ASA)Side Angle next (SAS)Side next Side (SSS)

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ASA theorem (Angle-Side-Angle)

The Angle next Angle Postulate (ASA) claims triangles are congruent if any kind of two angles and their consisted of side are equal in the triangles. An included side is the side in between two angles.

In the sketch below, we have △CAT and also △BUG. Notification that ∠C ~ above △CAT is congruent come ∠B on △BUG, and ∠A ~ above △CAT is congruent come ∠U top top △BUG.

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See the had side between ∠C and also ∠A on △CAT? that is equal in size to the contained side between ∠B and ∠U on △BUG.

The two triangles have actually two angle congruent (equal) and the had side in between those angles congruent. This forces the remaining angle on ours △CAT come be:

180° - ∠C - ∠A

This is due to the fact that interior angles of triangles include to 180°. You deserve to only do one triangle (or that is reflection) with given sides and angles.

You might think we rigged this, because we required you to look at certain angles. The postulate claims you have the right to pick any 2 angles and their included side. So walk ahead; look at at either ∠C and ∠T or ∠A and also ∠T on △CAT.

Compare them come the matching angles on △BUG. Girlfriend will watch that every the angles and all the sides room congruent in the two triangles, no issue which persons you pick to compare.

SAS organize (Side-Angle-Side)

By applying the Side Angle next Postulate (SAS), friend can likewise be certain your two triangles are congruent. Here, rather of picking two angles, we pick a side and its corresponding side on 2 triangles.

The SAS Postulate states that triangles are congruent if any kind of pair of equivalent sides and their consisted of angle room congruent.

Pick any side that △JOB below. An alert we room not forcing girlfriend to pick a specific side, because we know this functions no matter where you start. Move to the next side (in whichever direction you desire to move), which will move up an had angle.

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For the two triangles to it is in congruent, those three components -- a side, contained angle, and adjacent side -- should be congruent come the exact same three parts -- the matching side, angle and also side -- top top the other triangle, △YAK.

SSS to organize (Side-Side-Side)

Perhaps the most basic of the 3 postulates, Side next Side Postulate (SSS) says triangles are congruent if 3 sides the one triangle room congruent come the corresponding sides the the various other triangle.

This is the just postulate the does not address angles. You have the right to replicate the SSS Postulate using two directly objects -- uncooked spaghetti or plastic stirrers work-related great. Cut a tiny little off one, so it is not fairly as long as it began out. Cut the other length into two patent unequal parts. Currently you have actually three sides of a triangle. Placed them together. You have actually one triangle. Now shuffle the sides about and try to placed them together in a different way, to make a different triangle.

Guess what? you can"t do it. You deserve to only assemble your triangle in one way, no issue what girlfriend do. You deserve to think you are clever and also switch two sides around, but then every you have actually is a enjoy (a winter image) the the original.

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So when you establish that 3 lengths have the right to only make one triangle, you can see that two triangles with their 3 sides matching to each various other are identical, or congruent.


Checking Congruence in Polygons

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

Introducing a diagonal line into any of those shapes creates two triangles. Using any type of postulate, friend will find that the two developed triangles room always congruent.

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

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You currently know line SA, supplied in both triangles, is congruent to itself. What about ∠SAN? it is congruent come ∠WSA due to the fact that they are alternate interior angle of the parallel line segments SW and NA (because that the alternate Interior angle Theorem).

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You likewise know the line segment SW and also NA room congruent, since they were part of the parallelogram (opposite sides room parallel and also congruent).

So currently you have actually a next SA, an contained angle ∠WSA, and a side SW of △SWA. You have the right to compare those 3 triangle parts to the equivalent parts that △SAN:

Side SA ≅ Side SA (sure hope so!)Included edge ∠WSA ≅ ∠NASSide SW ≅ Side NA

Lesson Summary

After working your means through this lesson and giving it some thought, you currently are able to recall and apply 3 triangle congruence postulates, the next Angle next Congruence Postulate, Angle side Angle Congruence Postulate, and also the side Side side Congruence Postulate. You deserve to now determine if any two triangles room congruent!

Next Lesson:

Conditional Statements and also Their Converse


What you learned:

After you look over this lesson, check out the instructions, and also take in the video, you will be may be to:

Learn and also apply the Angle side Angle Congruence PostulateLearn and also apply the side Angle side Congruence PostulateLearn and also apply the side Side side Congruence Postulate