Triangle congruence: how to use SSS, ASA, and SAS
You don’t have to check all six parts to prove two triangles are congruent. Let’s see which three pieces are enough in each case, then use the markings to write the vertices in the right order.
What makes two triangles congruent?
Congruent triangles have the same size and shape. Their corresponding sides have equal lengths, and their corresponding angles have equal measures. The tick marks and arcs in a diagram tell us which parts are given as congruent.
SSS: Match all three sides
Side-side-side works when you know three pairs of corresponding sides have equal lengths. In this figure, AB matches CD, and AD matches CB. Both triangles use segment BD, so its length is the same in each triangle. That’s the reflexive property.
The markings pair with , with , and with . Keep those pairs in the same positions in the statement.
ASA: Match two angles and the side between them
In angle-side-angle, the included side connects the two marked angles. Here the arcs at A and D match. The marked angles at O are vertical angles, so they also have equal measures. The equal sides AO and DO connect the matched angles, giving us ASA.
SAS: Match two sides and the angle between them
In side-angle-side, the included angle is where the two marked sides meet. Here AC matches DC and BC matches EC. The marked angles at C are vertical angles, so the angle between those sides also matches.
Check the order in your congruence statement
Read the two names side by side. In the SAS example, A matches D, B matches E, and C matches C. You can start with a different vertex if you reorder both names together. Two sides and an angle away from where they meet don’t guarantee congruence in general, so check for that included angle before using SAS.
For another example, see how to prove triangles are similar.
