Similar Triangles

In this section we will be studying similar (~) triangles. It will be easier to remember these concepts if you start by considering size and shape.

Similar triangles have the same shape, but the size may be different.

Remember "@" means "is congruent to" and "~" is "similar to". Examples

Corresponding Triangles

Corresponding Equal Angles

Corresponding Proportional Sides

a/f = b/d = c/e = factor

DABC ~ DFDE

∠A @ ∠F

∠B @ ∠D

∠C @ ∠E

a/f = 6/3 = 2

b/d = 8/4 = 2

c/e = 10/5 = 3

DABC @ DFDE

∠A @ ∠F

∠B @ ∠D

∠C @ ∠E

a/f = 3/3 = 1

b/d = 4/4 = 1

c/e = 5/5 = 1

Two triangles are similar if:

two pairs of corresponding angles are equal (therefore the third pair of corresponding angles are also congruent).
the three pairs of corresponding sides are proportional.

Notice the corresponding angles for the two triangles in the applet are the same. The corresponding sides lengths are the same only when the scale factor slider is set at 1.0. Study the side lengths closely and you will find that the corresponding sides are proportional.

If you know triangles are similar, you can use the proportion of corresponding sides to help determine an unknown dimension.

Study the object below. You can change the triangles by dragging on the slider or dragging vertex A or B. The proportions of the corresponding pairs of sides changes as the scale slider position changes. Each time you stop dragging, look at the proportions. If one of the sides was unknown you could use two pairs of corresponding sides to calculate the missing dimension.