You also know that line segments SW and NA are congruent, because they were part of the parallelogram (opposite sides are parallel and congruent). 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 already know line SA, used in both triangles, is congruent to itself. Are they congruent? Checking Congruence in Polygons - Example Shape You now have two triangles, △SAN and △SWA. Suppose you have parallelogram SWAN and add diagonal SA. Using any postulate, you will find that the two created triangles are always congruent. Introducing a diagonal into any of those shapes creates two triangles. You can check polygons like parallelograms, squares and rectangles using these postulates. 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 think you are clever and switch two sides around, but then all you have is a reflection (a mirror image) of the original. You can only assemble your triangle in one way, no matter what you do. Now shuffle the sides around and try to put them together in a different way, to make a different triangle. Cut the other length into two distinctly unequal parts. You can replicate the SSS Postulate using two straight objects – uncooked spaghetti or plastic stirrers work great.Ĭut a tiny bit off one, so it is not quite as long as it started out. This is the only postulate that does not deal with angles. 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. Or 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. Move to the next side (in whichever direction you want to move), which will sweep up an included angle. Notice we are not forcing you to pick a particular side, because we know this works no matter where you start. The SAS Postulate says that triangles are congruent if any pair of corresponding sides and their included angle are congruent. Here, instead of picking two angles, we pick a side and its corresponding side on two triangles. SAS theorem (Side-Angle-Side)īy applying the Side Angle Side Postulate (SAS), you can also be sure your two triangles are congruent. 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. So go ahead look at either ∠C and ∠T or ∠A and ∠T on △CAT.Ĭompare them to the corresponding angles on △BUG. The postulate says you can pick any two angles and their included side. You may think we rigged this, because we forced you to look at particular angles. You can only make one triangle (or its reflection) with given sides and angles. This is because interior angles of triangles add to 180°. This forces the remaining angle on our △CAT to be:ġ80 ° − ∠ C − ∠ A 180°-\angle C-\angle A 180° − ∠ C − ∠ A The two triangles have two angles congruent (equal) and the included side between those angles congruent. 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. Notice that ∠C on △CAT is congruent to ∠B on △BUG, and ∠A on △CAT is congruent to ∠U on △BUG. In the sketch below, we have △CAT and △BUG. An included side is the side between two angles. The Angle Side Angle Postulate (ASA) says triangles are congruent if any two angles and their included side are equal in the triangles. Triangle Congruence Postulates and Theorems ASA theorem (Angle-Side-Angle) Let's take a look at the three postulates abbreviated ASA, SAS, and SSS. Testing to see if triangles are congruent involves three postulates. More important than those two words are the concepts about congruence. Do not worry if some texts call them postulates and some mathematicians call the theorems.
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