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Prove the HL Theorem without angle measures or      Pythagorean theorem. cannot include HL theorem in proof

Prove the HL Theorem without angle measures or      Pythagorean theorem. cannot include HL theorem in proof

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HL Theorem: If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

Triangle ABC Triangle DEF
Area of triangle ABC = AB x BC (For right-angle triangle area = base x height)
Area of triangle DEF = DE x EF  
If they are congruent the area is also the same.
AB x BC = DE x EF (One leg of the triangle is the same AB = DE, and the hypotenuse is also same )
BC = EF
Now, we have all three sides are the same of above two triangles. Now we can say they are congruent to each other.

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