Two shapes are similar when their corresponding angles are equal and their corresponding sides are all in the same ratio. Enlarging or shrinking a photograph produces a similar image: the shape is preserved, only the size changes. Because the angles must match exactly, checking angles is often the fastest way to prove two shapes are similar in the first place — if every angle in one triangle matches an angle in the other, the sides are automatically guaranteed to be in one consistent ratio.

Corresponding sides always sit opposite matching angles, so sketch and label both shapes' angles before deciding which sides pair up. This matters most when a diagram is drawn at an angle or one shape is rotated relative to the other — the "obviously matching" sides by position on the page are not always the true corresponding sides. Naming shapes consistently helps: in similar triangles $ABC$ and $XYZ$, the labelling itself tells you $AB$ corresponds to $XY$, $BC$ to $YZ$, and $CA$ to $ZX$, since vertices are listed in matching order.

Once $k$ is known from one pair of corresponding sides, multiply any other known length by $k$ to find its corresponding length on the similar shape:

Worked example. Triangles $ABC$ and $XYZ$ are similar, with $AB$ corresponding to $XY$ and $BC$ corresponding to $YZ$. Given $AB = 4.5\text{ cm}$, $XY = 9\text{ cm}$, $BC = 6.2\text{ cm}$, find $YZ$.

A classic exam context uses similar triangles formed by a vertical object and its shadow: the object, its shadow, and the sun's ray form one triangle; a shorter reference object (like a metre stick) and its own shadow, measured at the same moment, form a smaller similar triangle.

Worked example. A vertical flagpole casts a shadow $14\text{ m}$ long. At the same time, a $1.2\text{ m}$ stick standing nearby casts a shadow $2\text{ m}$ long. Find the height of the flagpole.

Since both shadows are measured at the same time, the sun's angle is identical, so the two right-angled triangles (object, shadow, ray) are similar: