Two triangles are congruent when one can be moved onto the other by rigid motions (slides, turns and flips) so that every side and every angle matches. That means six pairs of matching parts. The useful discovery is that you do not need to check all six. Three well-chosen pairs are enough, and knowing which three is the heart of triangle congruence in Geometry.
The idea behind the shortcuts
Congruent triangles have all corresponding sides and angles equal, and the reverse is also true: if all six pairs match, the triangles are congruent. The criteria below are shortcuts that guarantee the other three pairs will match too. Each one works because the given parts leave only one possible triangle. In the course, the criteria for ASA, SAS and SSS are explained from the definition of congruence in terms of rigid motions.
The four criteria that work
- SSS (side-side-side): three pairs of sides are equal. Example: both triangles have sides 5, 7 and 9. The sides fix all three angles, so the triangles match.
- SAS (side-angle-side): two pairs of sides and the angle between them. Example: AB = 6, AC = 8 and ∠A = 40° in one triangle; DE = 6, DF = 8 and ∠D = 40° in the other.
- ASA (angle-side-angle): two pairs of angles and the side between them. Example: angles of 50° and 60° with a 10-unit side joining them.
- AAS (angle-angle-side): two pairs of angles and a side that is not between them. This follows from ASA, because the angles in a triangle add to 180°. If two angles are 50° and 60°, the third is 70° in both triangles, so you can slide the known side into an ASA arrangement.
The word in the middle of each name tells you the position of the parts. In SAS and ASA the angle or side in the middle is the one that connects the other two given parts.
Using the criteria in a proof
Worked example 1. In quadrilateral ABCD, AB = CD and AD = CB. Show that triangle ABD is congruent to triangle CDB.
- AB = CD (given).
- AD = CB (given).
- BD = DB (the shared side, equal to itself).
- The triangles match by SSS, so ∠A = ∠C and the other matching parts are equal too.
Worked example 2. Segments AB and CD cross at M, and M is the midpoint of both. Show that AC = BD.
- AM = BM, because M is the midpoint of AB.
- CM = DM, because M is the midpoint of CD.
- ∠AMC = ∠BMD, because vertical angles are equal.
- So triangle AMC is congruent to triangle BMD by SAS.
- Corresponding parts of congruent triangles are equal, so AC = BD.
Step 5 is the real prize: once triangles are congruent, every remaining pair of corresponding parts is equal.
Why SSA does not work
Side-side-angle gives two sides and an angle that is not between them. It sounds like it should work, but it can produce two different triangles.
Worked example. Build a triangle with ∠A = 30°, AB = 10 and BC = 6, where BC is opposite ∠A. Draw a ray from A at 30° to AB. Point C must lie on this ray at distance 6 from B. The shortest distance from B to the ray is 10 × sin 30° = 5. Since 5 < 6 < 10, the circle of radius 6 around B crosses the ray in two points. So there are two triangles with the same SSA data, one with ∠C acute and one with ∠C obtuse. They are not congruent.
Because SSA can fail, it is not a congruence criterion. There is one special case: when the angle is a right angle (in a right triangle, the hypotenuse and a leg), the triangle is fixed. That version is called HL, and it is a special case you can show from the Pythagorean theorem.
Why AAA does not work either
Three pairs of equal angles guarantee only the same shape, not the same size. A triangle with angles 40°, 60° and 80° can be tiny or enormous. These are similar triangles, not congruent ones.
Common mistakes
- Matching parts in the wrong order. Write congruence statements so vertices correspond: if triangle ABC is congruent to triangle DEF, then A matches D, B matches E, C matches F.
- Using SAS when the angle is not between the sides. Check the position before naming the criterion.
- Claiming SSA. If your only evidence is two sides and a non-included angle, stop and look for a third piece of evidence.
- Forgetting free information. Shared sides, vertical angles and midpoints often supply the missing pair.
Practice Geometry questions with an explanation for every answer, or read the guide to California Geometry practice.
Source: the California Common Core State Standards for Mathematics, G-CO.7, G-CO.8.