Similarity of triangle
Similarity of triangle
In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional.
Video: Similar Triangles
It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent.
There are several statements each of which is necessary and sufficient for two triangles to be similar:
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The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is:
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If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar.
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All the corresponding sides have lengths in the same ratio:
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AB/A′B′ = BC/B′C′ = AC/A′C′. This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other.
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Two sides have lengths in the same ratio, .....
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