Similar Triangles

Same shape, different size

Two shapes are similar if they have the same shape but not necessarily the same size — one is simply a scaled copy of the other, blown up or shrunk down. We write \triangle ABC \sim \triangle DEF, and the symbol \sim is read "is similar to".

Being a scaled copy means two things hold together:

corresponding angles are equal — scaling never bends a corner, so the angles match exactly;
corresponding sides are in the same ratio — every side is multiplied by the same number, called the scale factor.

\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k

Here k is the scale factor: double everything and k = 2; halve everything and k = \tfrac12.

How to tell triangles are similar

As with congruence, you don't need to check everything — a few facts force the rest.

Two triangles are similar if any one of these holds:

AA — two angles of one equal two angles of the other. Since the three angles of a triangle always add to 180^\circ, matching two forces the third to match too, so the shapes line up.
SSS (in the same ratio) — all three pairs of corresponding sides share one common ratio k.
SAS — two pairs of sides are in the same ratio and the included angle between them is equal.

Congruence is just the special case where the scale factor is k = 1.

See the scaling

Step through a small triangle and a copy twice as big. The angles stay the same while every side doubles — so the two triangles are similar with scale factor k = 2.