Keeping a consistent correspondence

Book VI Definition 1 combines equal corresponding angles with proportional adjacent sides. The pairing must follow one consistent order around each polygon; exchanging a side pair to rescue one equation changes the comparison.

All rectangles have right angles, but a two-by-four rectangle is not similar to a two-by-six rectangle. Their aspect ratios differ. Equality of corresponding angles alone is insufficient for polygons in general.

References: [1]

The special case of triangles

VI.4 establishes proportional corresponding sides for triangles with equal corresponding angles. Its equiangular triangles need not each have three equal angles: the equality compares the two triangles.

Triangles with sides three, four, five and six, eight, ten share a scale factor of two. They are similar but not congruent. Changing just one side destroys that uniform proportional relationship.

References: [2]

Length and area scales

VI.19 relates similar-triangle areas to the square of the corresponding-side ratio. Tripling every length multiplies area by nine rather than three.

Conversely, equal area does not determine shape: a narrow triangle and a wider one can have the same area without matching angles or side ratios. Similarity, congruence and area equality answer different questions.

References: [3]