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Right Triangle Similarity Theorem

Rule

Right Triangle Similarity Theorem

Given a right triangle, if an altitude is drawn from the vertex of the right angle to the hypotenuse, then the two triangles formed are similar to the original triangle and to each other.

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According to this theorem, there are three relations that hold true for the diagram above.

Proof

Start by separating the two triangles formed by the altitude from
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By the Reflexive Property of Congruence, and Also, since all right angles are congruent, it is obtained that and
and and

Applying the Angle-Angle (AA) Similarity Theorem, it can be concluded that and are similar and and are similar. Then, by the Transitive Property of Congruence, and are also similar.

and