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Hypotenuse Leg Theorem

Proving Congruent Right Triangles

The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.


Example 1
$ \overline{AC} \cong \overline{ZY} \\ \overline{AB} \cong \overline{XZ} \\ m\angle ACB = m\angle ZYX = 90^{o} $
$ \triangle ACB \cong \triangle ZYX $
Statement Reason
$$ \angle ACB $$ and $$ \angle ZYX $$ are both right angles Given
$$ \triangle ABC$$ and $$ \triangle XYZ$$ are right triangles Right Triangles have a right angle
$$ \overline{AB} \cong \overline{XZ} $$ Given
$$ \overline{AC} \cong \overline{ZY} $$ Given
$$ \triangle ACB \cong \triangle ZYX $$ Hypotenuse Leg Theorem
Hypotenuse Leg Theorem
Example 2

The following proof simply shows that it does not matter which of the two (corresponding) legs in the two right triangles are congruent.

  • triangle symbol ABC and triangle symbol XZY are right triangles since they both have a right angle
  • AB = XZ (hypotenuse) reason: given
  • CB = XY (leg) reason: given
  • triangle symbol ABC congruent symbol triangle symbol XYZ by the hypotenuse leg theorem which states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.
Hypotenuse Leg Theorem Example
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