Similar Figures
Similar figures have the same shape but not necessarily the same size; corresponding angles are equal and corresponding sides are proportional.
Definition
Two shapes are similar if they have the same shape but possibly different sizes: all corresponding angles are congruent and all corresponding side lengths are proportional, sharing the same ratio $k$ (the scale factor), one is a scaled-up or scaled-down version of the other. For triangles, AA (two matching angles) is enough to prove similarity. Formally, two figures are similar if one can be mapped onto the other by a similarity transformation (a rigid motion composed with a dilation), an equivalence relation coarser than congruence; in the complex plane, similarity transformations are conformal maps of the form $f(z) = az + b$ (with $a$ nonzero), preserving angles and scaling lengths by $|a|$.
Example
A $3$-$4$-$5$ right triangle and a $6$-$8$-$10$ right triangle are similar, with every side of the second exactly twice the first, like a photograph enlarged or reduced. Triangle $ABC$ with sides $5$, $7$, $9$ and triangle $DEF$ with sides $10$, $14$, $18$ have ratio $2:1$ ($k=2$) and area ratio $k^2 = 4$, so the larger triangle has $4$ times the area of the smaller. All parabolas are similar (for example $y = x^2$ and $y = 4x^2$ are related by a scaling), and more surprisingly, all ellipses with the same eccentricity are similar too, a key result in projective geometry.
Key Insight
All squares are similar to each other, and all circles are similar to each other, but not all rectangles are similar (a $2 \times 3$ rectangle and a $2 \times 6$ rectangle share angles but not side ratios). When two figures are similar with scale factor $k$, their perimeters are in ratio $k$ and their areas are in ratio $k^2$, which is why enlarging a photo by factor $2$ doubles linear dimensions but quadruples the paper area needed. Fractal geometry is built on self-similarity, since a fractal looks similar to itself at every scale; the Sierpinski triangle is exactly similar to $3$ copies of itself scaled by $1/2$, and fractal dimension quantifies how similarity scales with complexity, generalizing the integer-valued notion of dimension.