SSS Congruence Rule

The SSS Congruence Rule states that if all three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

This means the triangles are identical in shape and size.

 Triangle 1:  a, b, c
 Triangle 2:  p, q, r
If a = p, b = q, c = r → the triangles are congruent.

SSS (Side-Side-Side) Rule:

If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

Symbolically, if in triangles \(\triangle ABC\) and \(\triangle DEF\):

• \(AB = DE\)
• \(BC = EF\)
• \(CA = FD\)

then:

\(\triangle ABC \cong \triangle DEF\)

When all three sides of a triangle are fixed, the triangle cannot change its shape. It is rigid. This means:

• You cannot form two different triangles with the same three side lengths.
• The angles automatically get fixed when the sides get fixed.

This is why matching only the sides is enough to prove congruence.

Imagine two triangles with the same side lengths: even if they appear rotated or flipped, they can still be exactly placed on top of each other.

      A                D
     / \              / \
    /   \            /   \
   B-----C          E-----F
 If AB=DE, BC=EF, CA=FD → congruent

• SSS checks only the sides; no angles are required.
• The order of writing triangles matters because matching must be consistent.
• SSS works for all types of triangles—scalene, isosceles, acute, right, or obtuse.

SSS is used wherever identical triangular pieces are needed:

• Triangles used in metal frames or towers,
• Matching roof truss components,
• Cutouts or templates used in construction,
• Geometric tiling patterns where triangle sides must align perfectly.