Triangle Congruence Theorems SSS, SAS, ASA, AAS, HL

What are Congruent Triangles? Congruent triangles are triangles that are the exact same shape and size. It's as if you put one in the copy machine and it spit out an identical copy to the one you already have. One might be rotated or flipped over, but if you cut them both out you could line them up exactly. They have three sets of sides with the exact same length and three sets of angles that have the same degree measure.

Often the triangles' angles will be marked with different numbers of arcs to indicate which angles have the same degree. In the diagram below, the angles marked with one arc each have the same angle measure, the angles with two arcs are matching and the angles marked with three arcs are matching. The sides will often be drawn with little tic marks to indicate sets of congruent sides. In the diagram above, the sides with one tic mark have the same length, the sides with two tic marks have the same length, and the sides with three tic marks have the same length.

How Do You Prove Two Triangles are Congruent? There are several different postulates you can use to prove that two triangles are congruent - that they are exactly the same size and shape. Side-Side-Side (SSS) Side-Angle-Side (SAS) Angle-Side-Angle (ASA) Angle-Angle-Side (AAS) Hypotenuse Leg (HL)

Included Sides and Angles In order to understand some of the ways to prove that triangles are congruent, it's important to understand the meaning of the word "included." Some of these postulates will say that the angle or side must be included - this word basically means in between, or in the middle.

An angle is included between the two sides that it touches. ​A side is included between the two angles it touches.

Side-Side-Side Postulate (SSS) The SSS postulate says that if all three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent. Let's say you have one triangle with side lengths of 3, 4, and 7 and a second triangle with side lengths of 3, 4, and 7. By the SSS Postulate, we know that the triangles must be exactly the same size and shape because it has three sets of congruent sides..

Side-Angle-Side Postulate (SAS) The SAS postulate says that if two sides of one triangle and the angle included between them are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. Let's say you have one triangle with side lengths 5 and 10 and the angle included between those two sides is 30 degrees. If you have a second triangle that also has side lengths 5 and 10 with a 30 degree angle in between, then by the SAS Postulate we know that the two triangles must be exactly the same size and shape. Make sure to notice the order of the letters in SAS. The A is in between the two S's. This indicates that the angle must be included between the two sides.

Angle-Side-Angle Postulate (ASA) The ASA Postulate says that if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. Let's say you have one triangle with angles that are 30 and 50 degrees and the side in between those two angles is 9 cm. If you have a second triangle with 30 and 50 degree angles and the side in between those angles is also 9 cm, then the triangles must be exactly the same shape and size by the ASA Postulate. Make sure to notice the order of the letters in ASA. The S is between the two A's. This indicates that the side must be included between the two angles.

Angle-Angle-Side Postulate (AAS) The AAS Postulate says that if two angles and the non-included side of one triangle are congruent to two angles and the non-included side of a second triangle, then the triangles are congruent. This is very similar to the ASA Postulate because it also has two angles and a side. However, the big important difference between the two is where the side is located. In ASA, the side is in between the two angles on the triangle. In AAS, the S is listed at the end to indicate that the side is NOT in between the angles. It doesn't matter which side is marked, as long as it's not the side that's between the two angles.  In the diagram, the bottom two sides could have been marked instead.

Hypotenuse-Leg (HL) If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. Recall that the criteria for our congruence postulates have called for three pairs of congruent parts between triangles. The HL Theorem essentially just calls for congruence between two parts: the hypotenuse and a leg.

What about AAA? Let's start with AAA - why isn't there an AAA Postulate? AAA would mean that you have triangles with 3 sets of congruent angles. Would these two triangles have to be exactly the same size and shape? Unfortunately not. We can draw two triangles with the exact same angles and make one larger than the other. They'll be the same shape, but not the same size. This means that AAA cannot​ be used to prove that triangles are congruent.

What about SSA? If two sides and the non-included angle are congruent, are the triangles congruent? No, and it's not just because SSA spells a foul word backwards. SSA just doesn't work to prove that two triangles are congruent. ​ It's possible to draw two different triangles with these requirements that are not congruent. Take a look at the two triangles. They have two sets of congruent sides and a non-included angle. See how the side on the right can be rotated out to form a larger triangle? This is why SSA doesn't guarantee that two triangles will be the same. When the angle is included between the two sides (in SAS), it's as if the top angle gets locked into place to prevent this from happening.

Angle, Angle, Side - AAS

Angle, Side, Angle - ASA

Hypotenuse Leg - HL

Side, Side, Side - SSS

Side, Angle, Side - SAS