SSS Similarity Rule

The SSS Similarity Rule states that if the three sides of one triangle are in the same ratio as the corresponding three sides of another triangle, then the triangles are similar.

This means their shapes are identical even if their sizes differ.

AB / DE = BC / EF = CA / FD → Triangles are similar

SSS (Side-Side-Side) Similarity Rule:

If in triangles \(\triangle ABC\) and \(\triangle DEF\):

\( \dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{CA}{FD} \)

then:

\( \triangle ABC \sim \triangle DEF \)

If all three pairs of corresponding sides are in the same ratio:

• The triangles expand or shrink in the same proportion.
• No angle can change without affecting the side ratios.
• The triangles must have identical angle measures.

This makes the shapes identical even if the triangles are not the same size.

        A                          D
       / \                        / \
      /   \                      /   \
     B-----C                    E-----F
AB/DE = BC/EF = CA/FD → Similar Triangles

• SSS checks only side ratios — no angles are needed initially.
• Triangles may be rotated or flipped but still similar.
• SSS works for all triangle types: scalene, isosceles, acute, obtuse, and right.

SSS similarity is used when only side lengths are available, such as:

• Scaling models in architecture and engineering,
• Proportion-based problems in surveying,
• Determining unknown distances using ratios,
• Checking if two triangular components are proportionally identical.