Similarity of Shapes

Conditions for Triangle Similarity

If two triangles and are similar, then:
  1. The ratios of the lengths of their corresponding sides are equal.
  2. The measures of their corresponding angles are equal.
This is illustrated in the figure below:
Conversely, if the ratios of the lengths of the three pairs of corresponding sides are equal and the measures of the three pairs of corresponding angles are equal, then the two triangles are similar. However, it is not always necessary to check all these conditions. In most cases, fulfilling one of the following simplified conditions is sufficient to confirm that two triangles are similar.

Triangle Similarity Conditions

Two triangles are similar if they satisfy any one of the following conditions. These are called the conditions for triangle similarity:
  1. SSS Similarity: The ratios of the lengths of all three pairs of corresponding sides are equal:
  1. SAS Similarity: The ratios of the lengths of two pairs of corresponding sides are equal, and the included angles are equal:
  1. AA Similarity: Two pairs of corresponding angles are equal: