Triangles [Congruency in Triangles]
212. Three Sides, One Shape · SSS Congruency Theorem
If all three corresponding sides are equal, the triangles are congruent.
By the SSS rule (Side–Side–Side), if the three sides of one triangle equal the three sides of another, the triangles are congruent. Three fixed side lengths can close into only one triangular shape, leaving no freedom for the angles, which is why a triangle is a rigid figure.
What this lesson covers
Try to break it
Drag C and F. Whenever the three sides of △ABC match those of △DEF in length, the triangles come out the same shape. Try to find a position where the sides match but the shapes don't — impossible. Once the three side lengths are fixed, there's only one triangle (up to flipping).
How you build it
Construct a triangle from three sides (SSS).
- Segment tool: draw base DE — any length.
- Circle tool: centre D, drag out to your chosen length for side DF.
- Circle tool: centre E, drag out for side EF — it meets the first circle at F.
- Point tool: mark F where the two circles cross.
- Segment tool: join D to F.
- Segment tool: join E to F. Three fixed sides make exactly one triangle — that is SSS.
The proof, step by step
Prove that three equal sides make the triangles congruent (SSS).
- In ΔABC and ΔDEF, AB = DE, BC = EF, and AC = DF (Given)
- Place ΔABC on ΔDEF such that A coincides with D and side AB lies along DE.
- Since AB = DE, point B coincides exactly with point E.
- Point C must coincide with F because AC = DF and BC = EF. There is only one possible position for C.
- Therefore, ΔABC ≅ ΔDEF by SSS Congruence Criterion.
Worked example
In ΔABC and ΔDEF, AB = 5 cm, BC = 6 cm, AC = 7 cm and DE = 5 cm, EF = 6 cm, DF = 7 cm. Which congruence criterion proves ΔABC ≅ ΔDEF?
Since all three corresponding sides are equal (AB=DE, BC=EF, AC=DF), the SSS congruence criterion applies directly. No angles are needed.
- SAS (Side-Angle-Side)
- ASA (Angle-Side-Angle)
- SSS (Side-Side-Side) — correct
- RHS (Right-Hypotenuse-Side)