Congruent Triangles: SSS, SAS, ASA, AAS and HL
Two triangles are congruent if they have exactly the same shape and size, and several short checklists (SSS, SAS, ASA, AAS and HL) prove it without comparing all six parts. In brief: SSS needs three pairs of equal sides, SAS two sides and the angle between them, ASA two angles and the side between them, AAS two angles and a side not between them, and HL (for right triangles) the hypotenuse and one leg. The pattern SSA is not a valid criterion in general, and AAA gives only similarity.
Congruent triangles allow you to conclude that all corresponding parts are equal, which is the basis of most geometric proofs. This guide states each criterion, shows why the invalid patterns fail, gives worked examples including a short proof and explains how to write the correspondence of vertices correctly.
What does congruent mean?
Triangles ABC and DEF are congruent, written △ABC ≅ △DEF, if one can be moved onto the other by a translation, rotation or reflection. Then the three pairs of corresponding sides and the three pairs of corresponding angles are equal. The order of the letters matters: △ABC ≅ △DEF means A corresponds to D, B to E and C to F, so AB = DE, BC = EF, CA = FD, ∠A = ∠D and so on.
The principle “corresponding parts of congruent triangles are congruent” (CPCTC) is used in proofs after congruence has been established.
What are the congruence criteria?
SSS: side-side-side
If three sides of one triangle equal three sides of the other, the triangles are congruent. A triangle is determined by its three sides, since the construction gives a unique shape up to reflection.
SAS: side-angle-side
If two sides and the included angle (the angle between them) are equal, the triangles are congruent.
ASA: angle-side-angle
If two angles and the included side are equal, the triangles are congruent.
AAS: angle-angle-side
If two angles and a non-included side are equal, the triangles are congruent. This follows from ASA, since the third angle is determined by the angle sum, which makes the included side available.
HL: hypotenuse-leg (right triangles only)
If two right triangles have equal hypotenuses and one pair of equal legs, they are congruent. The other leg follows from the Pythagorean theorem, and then SSS applies.
Which patterns do not work?
- AAA: three equal angles give triangles of the same shape but not necessarily the same size. They are similar. A triangle with angles 60°, 60°, 60° can have any side length.
- SSA (or ASS): two sides and a non-included angle can give two different triangles. Example: a side of 8, a side of 5 opposite a 30° angle: the second side can be placed in two positions, giving two triangles with the same given data. The exception is a right angle (HL) or when the given angle is opposite the longer side.
The ambiguous case is also met in the law of sines.
How do you use the criteria?
Example 1: choosing a criterion
Triangle ABC has AB = 7, BC = 9 and ∠B = 40°. Triangle DEF has DE = 7, EF = 9 and ∠E = 40°. Two sides and the included angle are equal: SAS, so △ABC ≅ △DEF with A ↔ D, B ↔ E, C ↔ F. Therefore AC = DF.
Example 2: two triangles sharing a side
In a quadrilateral ABCD with AB = CD and BC = DA, show that ∠B = ∠D. Draw the diagonal AC. Triangles ABC and CDA have AB = CD, BC = DA and the common side AC. By SSS, they are congruent, and by CPCTC ∠B = ∠D.
Example 3: a proof with the angle bisector
Given: AD bisects ∠A in triangle ABC and AD ⟂ BC. Prove: AB = AC. In triangles ABD and ACD, ∠BAD = ∠CAD (given), AD = AD (common) and ∠ADB = ∠ADC = 90° (given). This is ASA, so the triangles are congruent, and AB = AC.
Example 4: HL
Two right triangles have hypotenuses 13 and legs 5. The other legs are both √(169 − 25) = 12, and they are congruent by HL (or SSS).
Example 5: counterexample for SSA
Construct a triangle ABC with ∠B = 30°, BA = 8 and AC = 5. The distance from A to the line BC is 8 · sin 30° = 4. Because 4 < 5 < 8, the circle of radius 5 around A meets the line BC in two points, C1 and C2. Both triangles ABC1 and ABC2 have the same given parts (SSA) but are not congruent: BC1 ≈ 3.9 and BC2 ≈ 9.9. If AC were 4, there would be one right triangle, and if AC were 8 or more, only one triangle.
Summary table of criteria
| Criterion | Given equal parts | Valid? |
|---|---|---|
| SSS | 3 sides | Yes |
| SAS | 2 sides and the included angle | Yes |
| ASA | 2 angles and the included side | Yes |
| AAS | 2 angles and a non-included side | Yes |
| HL | hypotenuse and a leg of right triangles | Yes |
| AAA | 3 angles | No (only similar) |
| SSA | 2 sides and a non-included angle | No (in general) |
Where do the criteria come from?
Euclid proved SAS, ASA and SSS in Book I of the Elements (Propositions 4, 26 and 8). His proof of SAS uses the idea of placing one triangle on top of the other, which later critics found to rely on an unstated assumption about motion. In many modern axiom systems, SAS is taken as an axiom. Hilbert’s axioms of 1899 include it for this reason.
Where is congruence used today?
Congruence underlies measurement by triangulation and the correctness of manufacturing: identical parts must be congruent. Structural frames made of triangles keep their shape because a triangle with given side lengths is rigid, which is the SSS criterion in practice. Proof checking and computer vision use congruence to match shapes.
Which mistakes are common?
Using SSA as a criterion
Two sides and a non-included angle do not guarantee congruence, except in a right triangle (HL).
Writing a wrong correspondence
Write △ABC ≅ △DEF with matching letters in matching positions. A wrong order gives false conclusions.
Confusing congruence with similarity
AAA proves similarity, not congruence.
Assuming equality from the picture
Use only marked or proved equalities, not lengths that seem equal in the diagram.