HL (Hypotenuse-Leg) Congruence

For two right triangles, if the hypotenuse of one right triangle is congruent to the hypotenuse of the other, and if one leg of one right triangle is congruent to one leg of the other, then the two triangles are congruent.

Suppose that the hypotenuse is $h$, and the common leg of the two triangles is $x$. Let the remaining legs of the triangles be $y_1$ and $y_2$.

Whenever one thinks of right triangles, the Pythagorean theorem should immediately come to mind.

Using the Pythagorean Theorem on both triangles, $$x^2+y_1^2=h^2$$ $$x^2+y_2^2=h^2$$ Subtracting the two equations, $$y_1^2=y_2^2$$ Since undirected Euclidean distances are always nonnegative, $$y_1=y_2$$ Now, by SSS congruence, the two triangles are congruent.

We have thus proved that the HL congruence is a valid method of seeing whether two right triangles are congruent.

Notice that the HL congruence is a special case of SSA. The main reason that SSA failed is that the equation $\sin\theta=x$ actually has two solutions for $\theta$ on the domain $[0^\circ, 180^\circ]$. Thus, there would be two possible locations for the last vertex. However, notice that the right angle has a unique sine; no other angle in this domain has a sine of $1$. Thus, the HL congruence succeeds while the SSA congruence fails.

Here is a great triangle drawing tool that can help you out in this topic.