Why HL Works

HL is SSS in disguise. In a right triangle the Pythagorean theorem ties the three sides together: leg² + leg² = hypotenuse². So once the hypotenuse and one leg are known, the other leg is fixed.

Here ${5}^{2}+{b}^{2} = {13}^{2}$, so ${b}^{2} = 169-25 = 144$. Every right triangle with hypotenuse 13 and a leg of 5 has its other leg equal to 12. Two such triangles have three equal pairs of sides, and SSS makes them congruent. That is why HL can skip the third side.