Similar Triangles

Two triangles are similar if corresponding angles are equal and corresponding sides have the same ratio.

Example:

Let's spell this out carefully:

There is a correspondence between the vertices of the two triangles ABC and A'B'C', with A corresponding to A', and B corresponding to B', and C corresponding to C'.

(When one says "triangle ABC is similar to triangle XYZ," it is usually understood that the vertices correspond in the given order. In other words, A corresponds to X, and B corresponds to Y, and C corresponds to Z.)

Corresponding angles are equal:

  • angle A = angle A'
  • angle B = angle B'
  • angle C = angle C'

(The signs of angles can be taken into account. We'll discuss this later.)

The ratios of corresponding sides are equal:

There is a positive number r such that

  • A'B' / AB = r
  • B'C' / BC = r
  • A'C' / AC = r

This number r is called the similarity ratio of triangle A'B'C' to triangle ABC.

Thus the similarity of two triangles involves 6 equations, three about angles and two about ratios of sides.

Another way to express the equality of ratios of sides is to say

  • AB / BC = A'B' / B'C'
  • BC / AC = B'C' / A'C'
  • AC / AB = A'C' / A'B'

There are several important theorems about similar triangles. Each of these theorems states that if some of the 6 equations hold, then all 6 of the equations must hold; in other words, the triangles must be similar.

Side-Angle-Side (SAS) theorem: If two triangles have one pair of corresponding angles equal and both corresponding pairs of sides adjacent to the angle have the same ratio, then the triangles are similar. In other words, if triangles ABC and A'B'C' have angle A = angle A' and AB / A'B' = AC / A'C', then the triangles are similar.

Angle-Angle (AA) theorem: If two triangles have two pair of corresponding angles equal, then the triangles are similar. In other words, if triangles ABC and A'B'C' have angle A = angle A' and angle B = angle B', then the triangles are similar.

Side-Side-Side (SSS) theorem: If two triangles have all three pairs of corresponding sides in the same ratio, then the triangles are similar. In other words, if triangles ABC and A'B'C' have A'B' / AB = B'C' / BC = C'A' / CA , then the triangles are similar.

But the "Angle-Side-Side (ASS) theorem" is not true.

Example:

If the triangles ABC and A'B'C' are similar, and the similarity ratio of A'B'C' to ABC is r, then the ratio of the area of A'B'C' to the area of ABC is r2.