similar triangles
Similar triangles are two triangles with the same shape but possibly different sizes — like a small flag and a giant version of the same flag. Their corresponding angles are equal and their corresponding sides are in the same ratio. Because a triangle is so rigid, this is by far the most useful instance of similarity in all of geometry, and most 'find the missing length' problems eventually reduce to spotting a pair of similar triangles.
If triangle ABC ~ triangle DEF, then angle A = angle D, angle B = angle E, angle C = angle F, and the sides match in proportion: AB/DE = BC/EF = CA/FD. That single common value is the scale factor. The letter order in the similarity statement is a dictionary: A corresponds to D, B to E, C to F, so you always pair the sides that lie between matching pairs of vertices. Once you know two triangles are similar, you can set up a proportion from any pair of corresponding sides and solve for an unknown length.
Triangles are special because you do not have to check everything. Thanks to the angle-sum being 180 degrees, knowing two angles match (AA) already forces the third to match and the sides to be proportional — so similarity of triangles can be established far more cheaply than similarity of general figures. This is why surveyors, photographers, and astronomers reach for similar triangles whenever they need to measure something they cannot reach.
A 2 m stick casts a 3 m shadow; at the same moment a flagpole casts a 24 m shadow. The sun's rays make the two right triangles similar, so 2/3 = h/24, giving h = 16 m for the pole.
Two triangles sharing the sun's angle are similar; one proportion finds the unreachable height.
Be strict about correspondence. Two triangles can be similar even though their drawings are flipped or rotated; you must match angles to angles, not just write the vertices in the order they happen to be labelled. Pairing the wrong sides is the most common source of a wrong answer.