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The Hypotenuse-Leg (HL) Theorem says that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent. HL looks like the side-side-angle pattern, which does not prove congruence in general. It works here because the right angle, together with the Pythagorean theorem, fixes the length of the third side, which reduces HL to SSS.
Have students try to build two different right triangles with a hypotenuse of 13 and a leg of 5; they will find the other leg is forced to be 12. Then contrast with a general SSA example where two different triangles fit the same data. Practice proofs where a shared side or a perpendicular creates the right triangles, such as showing that the two tangent segments from an outside point to a circle are equal.
You have learned SSS, SAS, ASA and AAS as ways to prove two triangles congruent. You have also learned that SSA (two sides and an angle that is not between them) does not work in general. With an acute angle, a fixed side next to it, and a fixed side opposite it, the opposite side can often swing into two different positions, giving two different triangles.
There is one important exception. Suppose both triangles are right triangles, and you know that their hypotenuses are congruent and that one pair of legs is congruent. This is still a side-side-angle pattern, since the right angle is not between the hypotenuse and the given leg. Yet the triangles must be congruent. This is the Hypotenuse-Leg Theorem, usually written HL.
The reason comes from the Pythagorean theorem. In a right triangle with hypotenuse c and legs a and b, we have a² + b² = c². If you know c and a, then b = √(c² - a²), and there is only one positive value. So the third side is determined. Two right triangles with the same hypotenuse and the same leg therefore have all three sides equal, and SSS finishes the proof. For example, any right triangle with hypotenuse 13 and leg 5 has other leg √(169 - 25) = √144 = 12.
To use HL in a proof, you must establish three things: both triangles have a right angle, the hypotenuses are congruent, and one pair of legs is congruent. Watch for right angles that come from perpendicular lines, altitudes, or a tangent meeting a radius.
A classic application involves a circle. Draw tangent segments PA and PB from an outside point P, and draw radii OA and OB. A tangent is perpendicular to the radius at the point of tangency, so triangles OAP and OBP are right triangles. They share the hypotenuse OP, and OA = OB because both are radii. By HL the triangles are congruent, so PA = PB. Two tangent segments drawn to a circle from the same outside point are always equal in length.
Keep HL separate from the case of two congruent legs. Two legs with the right angle between them is an SAS situation, so it needs no special theorem.