Similarity

Two geometric figures are similar if one is a scaled (and possibly rotated/reflected) copy of the other. Notation: $\triangle ABC \sim \triangle DEF$.

Conditions for similarity (triangles):

• AA: two pairs of equal angles → similar (the third pair must match because angles sum to $180°$).
• SAS: two pairs of proportional sides + equal included angle → similar.
• SSS: three pairs of proportional sides → similar.

Key consequences:

• All corresponding angles are equal.
• All corresponding sides are proportional with the same ratio $k$ (the scale factor).
• Areas scale by $k^2$, volumes scale by $k^3$.

Similarity is the foundation of:

• Trigonometry — the trig ratios depend only on angle, not triangle size, because all right triangles with the same angle are similar.
• Map scales and architectural drawings.
• Fractals and self-similar structures.
• Image scaling in graphics — preserves visual identity by being a similarity transformation.

Distinguish from congruence: congruent means similar and equal in size (scale factor 1).