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.

Start by separating the two triangles formed by the altitude

\overline{C D}

from

△ A B C .

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By the Reflexive Property of Congruence,

∠ B ≅ ∠ B

and

∠ A ≅ ∠ A .

Also, since all right angles are congruent, it is obtained that

∠ B D C ≅ ∠ B C A

and

∠ C D A ≅ ∠ B C A .

$△ C B D$ and $△ A B C$	$△ A C D$ and $△ A B C$
$∠ B ≅ ∠ B$	$∠ A ≅ ∠ A$
$∠ B D C ≅ ∠ B C A$	$∠ C D A ≅ ∠ B C A$

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

$△ C B D \sim △ A B C$ and $△ A C D \sim △ A B C$ $⇓ △ C B D \sim △ A C D$