Two triangles are considered to be similar if for each angle in one triangle, there is a congruent (that is, equivalent) angle in the other triangle. For example, these two triangles are similar:

A
|\
| \         ΔABC ~ ΔDEF
|  \
|   \    D
|    \   |\
|_____\  |_\
B      C E  F

Similarity is a valuable concept in geometry and trigonometry because the lengths of the corresponding sides of two similar triangles are proportional to each other. Triangles with proportional sides have equivalent angles, and vice-versa. So all 30-60-90 triangles are similar, all equilateral triangles are similar, all triangles with sides proportional to 3, 4, and 5 are similar, and so on. Two similar triangles are scaled representations of each other. Note that two triangles can be rotated or flipped with respect to each other and still be similar.

If two triangles are similar, such as the above, we know the following:

∠A ≅ ∠D (Angle A is congruent to angle D)
∠B ≅ ∠E
∠C ≅ ∠F
AB ∝ DE (Line segment AB is proportional to line segment DE)
BC ∝ EF
AC ∝ DF

 AB     BC     AC
---- = ---- = ----
 DE     EF     DF
    Testing for Similarity:
  1. Angle-Angle Test
    if two corresponding angles on the triangles are equivalent, then the third pair must also be equivalent, so the triangles are similar.
  2. Side-Angle-Side Test
    if two corresponding sides are proportional to each other and the angle between each of those pairs of sides are equivalent, then the triangles are similar.
  3. Side-Side-Side Test
    if each side of a triangle has a corresponding proportional side on another triangle, the triangles are similar.
Test one is sufficient because, since the angles of all triangles add up to 180 degrees, knowing two angles defines the third. Tests two and three rely on the fact that you can only make one triangle out of a given three lengths for sides.

Similar triangles prove themselves valuable when dealing with ratios or proportions. For example, if you wanted to measure the height of a 100-storey building, it would be impractical to use a 1000-foot tape measure. One method for using similar triangles to measure the building would be to use a 12-inch ruler to create a kind of "scale model" of the building. Standing the ruler perpendicular to the ground, measure the length of its shadow, and then measure the length of the building's shadow. Since the shadows are falling at the same angle, the triangles formed by the building and the ruler are similar. The height of the building is proportional to the height of the ruler at the same ratio as the length of the shadows.

Example:
A building casts a 21 foot shadow along the ground. A 12 inch ruler casts a 3 inch shadow. How tall is the building?
The building's height is AB, and the ruler's height is DE. The building's shadow is BC, and the ruler's shadow is EF. Similar triangles tells us that AB/DE = BC/EF, and we want to find AB. So rearrange the equation: AB = DE × BC/EF. Convert the ruler's dimensions to feet (1 foot ruler, 0.25 foot shadow) and solve.

       21'
1' x ------- = 84'   The building is 84' tall.
      0.25'

Applications:
Similar triangles can be used in many situations in which angles of two differently-sized triangles are the same. Optics, scale modeling, trigonometry, surveying, astronomy, and many, many other applications of mathematics rely on the concept of similarity.

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