Similar Triangles

Definition: Triangles are similar if they have the same shape, but can be different sizes. (They are still similar even if one is rotated, or one is a mirror image of the other).

Try this Drag any orange dot at either triangle's vertex. Both triangles will change shape and remain similar to each other.

Triangles are similar if they have the same shape, but not necessarily the same size. You can think of it as "zooming in" or out making the triangle bigger or smaller, but keeping its basic shape. In the figure above, as you drag any vertex on triangle PQR, the other triangle changes to be the same shape, but half the size. In formal notation we can write

which is read as "Triangle PQR is similar to triangle P'Q'R' ". The letter with a small vertical dash after it such as P' is read as "P prime".

Properties of Similar Triangles

1. Corresponding angles are the congruent (same measure)

So in the figure above, the angle P=P', Q=Q', and R=R'.

2. Corresponding sides are all in the same proportion

Above, PQ is twice the length of P'Q'. Therefore, the other pairs of sides are also in that proportion. PR is twice P'R' and RQ is twice R'Q'. Formally, in two similar triangles PQR and P'Q'R' :

Rotation One triangle can be rotated, but as long as they are the same shape, the triangles are still similar. In the figure below, the triangle PQR is similar to P'Q'R' even though the latter is rotated clockwise 90.

In this particular example, the triangles are the same size, so they are also congruent.

Reflection One triangle can be a mirror image of the other, but as long as they are the same shape, the triangles are still similar. It can be reflected in any direction, up down, left, right. In the figure below, triangle PQR is a mirror image of P'Q'R', but is still considered similar to it.

How to tell if triangles are similar Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two triangles are similar. Various groups of three will do. Triangles are similar if:

1. AAA (angle angle angle) All three pairs of corresponding angles are the same. See Similar Triangles AAA.

2. SSS in same proportion (side side side) All three pairs of corresponding sides are in the same proportion See Similar Triangles SSS.

3. SAS (side angle side) Two pairs of sides in the same proportion and the included angle equal. See Similar Triangles SAS.

Similar Triangles can have shared parts Two triangles can be similar, even if they share some elements. In the figure below, the larger triangle PQR is similar to the smaller one STR. S and T are the midpoints of PR and QR respectively. They share the vertex R and part of the sides PR and QR. They are similar on the basis of AAA, since the corresponding angles in each triangle are the same.

identity An equation which states that two expressions are equal for all values of any variables that occur, such as x2 y2 = (x + y)(x y) and x(x 1)(x 2) = x3 3x2 + 2x. Sometimes the symbol is used instead of = to indicate that a statement is an identity. identity function The function which maps every element to itself, i.e. I(x) = x for every x. itself, i.e. I(x) = x for every x.

Similar TrianglesProperties of Similar Triangles1. Corresponding angles are the congruent (same measure)2. Corresponding sides are all in the same proportion

RotationReflectionHow to tell if triangles are similarSimilar Triangles can have shared parts