We say that two triangles are congruent if one can be moved, rotated, and possibly reflected to lie on top of each other exactly. You can think of congruence as a more precise way of saying that two triangles are "the same". Here is an example to illustrate:
Here we are told by the markings on each triangle that the three sides are equal, and the three angles are equal as well - this is the given information. With a reflection, a rotation, and some translation, we will be able to place these triangles directly on top of each other.
Most of the time we are not given all three sides and all three angles. So the big question is: what is the minimum amount of information we need to know about two triangles to conclude that they are congruent? It turns out there are five different sets of information, all of which are sufficient to demonstrate congruence.
If we know that each side of one triangle has a matching congruent side in the other triangle, then the triangles must be congruent. You can try this yourself with three straight objects; you can only make one triangle without changing the lengths, remembering that we are counting mirror images as being congruent as well:
This kind of congruence is called side-side-side, or just SSS.
If we know that two triangles have a pair of matching sides, and the angles between each pair are congruent, then the triangles must be congruent. You can try this yourself with any two straight objects - if you hold them together at a point and a certain angle apart along the rest of their lengths, there is only one triangle you can form by joining the ends together:
This kind of congruence is called side-angle-side, or just SAS.
It is possible to have two triangles with a matching pair of sides and a matching angle that are not congruent, like these two:
Try using this applet to find the two different triangles with two congruent sides and a congruent angle, just like the picture above:
This congruence test also uses two sides, and is an exception to the "the congruent angle must be between two congruent sides" rule. However, it only applies to right triangles. If two right triangles have congruent hypotenuses and one pair of legs are congruent, then the triangles must be congruent overall.
This congruence test is called hypotenuse-leg, or just HL.
What if we only know that one side is congruent? In this case we need to know that there are two pairs of congruent angles. There are different cases that we need to treat separately, the first being when the two given angles lie on either end of the given side:
This kind of congruence is called angle-side-angle congruence, or just ASA.
The other way to use a single pair of congruent sides side to prove congruence of triangles is to know one pair of angles on the congruent sides are equal, and that the angles opposite the congruent sides are also equal:
This kind of congruence is called angle-angle-side congruence, or just AAS.
Here is a summary of the five triangle congruence tests.
To show that two triangles are congruent, it is sufficient to demonstrate the following:
Consider the following three triangles:
Which of the following triangles are congruent?
State the reason why the two previous triangles are congruent:
Consider the given triangles.
Do we have enough information to deduce that the two triangles are congruent?
Consider the adjacent figure:
From the information given on the diagram, which angle is congruent to $\angle PSQ$∠PSQ?
State the most direct reason why $\triangle PSQ$△PSQ is congruent to $\triangle RQS$△RQS.