Same Shape, Any Size Triangles (Similarity)

All circles are similar to each other, and all squares are similar to each other too -- but general polygons need TWO separate conditions to count as similar, not just one: their corresponding angles must be equal, AND their corresponding sides must be in the same proportion. Neither condition alone is enough -- a square and a rectangle can share all four right angles without being similar (their sides aren't proportional), and a square and a rhombus can share four equal sides without being similar either (their angles don't match).

The Basic Proportionality Theorem, credited to Thales, states that a line drawn parallel to one side of a triangle, cutting the other two sides at distinct points, always divides those two sides in exactly the same ratio. The proof compares the areas of the triangles this parallel line creates: joining BE and CD and dropping the two relevant heights shows that triangle ADE's area over triangle BDE's area equals AD over DB (same height from E), while triangle ADE's area over triangle DEC's area equals AE over EC (same height from D) -- and since triangles BDE and DEC share the exact same base DE and sit between the same two parallel lines, their areas are exactly equal, which forces AD over DB to equal AE over EC directly.

Just as with congruence, checking every single angle and every single side to prove similarity would be exhausting -- so three shortcut criteria do the same job with far less information. The AA (or AAA) criterion says matching just two angles is enough, since the third angle is then automatically forced to match too (angles of a triangle always sum to 180 degrees), and matching angles always force proportional sides. The SSS similarity criterion says proportional sides alone are enough, forcing the angles to match. The SAS similarity criterion says one matching angle plus its two adjacent sides being proportional is enough. All three are proved with the same underlying construction: marking off a smaller triangle inside the larger one that exactly matches the given information, and showing it must be congruent to (or a scaled copy of) the original.

A girl 90 cm tall, walking away from a 3.6 m lamp-post at 1.2 metres per second, has her shadow's length grow in a way that similar triangles predict exactly: after 4 seconds she has walked 4.8 m from the post, and comparing the similar triangles formed by the lamp-post-and-full-shadow versus the girl-and-her-own-shadow gives (4.8 plus the shadow length) over the shadow length equals 3.6 over 0.9, which solves out to a shadow exactly 1.6 m long.