The Proof of the Hypotenuse Leg Theorem

The Hypotenuse Leg Theorem states that if the hypotenuse and one leg of a triangle are congruent to the hypotenuse and leg of another triangle, then the two triangles are congruent.

hypotenuse leg theorem

In the figure above, ABC and DEF are right triangles with right angles at C and F and with AB \cong DE and BE \cong EF. We are going to show that the two triangles are congruent. 

Given

Triangle ABC and DEF, right angled at C and F respectively, with AB \cong DE and BC \cong EF.

Proof

It is given that AB \cong DE, BC \cong EF, so by the definition of congruence,

AB = DE and BC = EF.

Now, by Pythagorean Theorem,

AB^2 = AC^2 + BC^2 and DE^2 = DF^2 + EF^2.

Since DE = AB, by substitution, we have

AC^2 + BC^2 = DF^2 + EF^2

Now, since BC = EF,

AC^2 + EF^2 = DF^2 + EF^2

Subtracting EF^2 from both sides, we have

AC = DF.

So, by the SSS Congruence, ABC \cong DEF.

 

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