HL Congruence for Right Triangles
Two sides and a non-included angle — SSA — is famously not a congruence criterion, because that information can build two different triangles. But there is one special case where it works: when the angle is a right angle. That case gets its own name, HL, for Hypotenuse-Leg.
HL has exactly three requirements: both triangles are right triangles, their hypotenuses are congruent, and one pair of corresponding legs is congruent. Check all three and the triangles are congruent — skip any one and HL does not apply.
What HL requires
First, both triangles must be right triangles — HL never applies anywhere else. Second, the H: the hypotenuses (the sides opposite the right angles, always the longest sides) must be congruent. Third, the L: one pair of corresponding legs must be congruent.
In a diagram, look for the small squares marking the right angles, then check the tick marks. One matched pair of ticks on the hypotenuses plus one matched pair on a leg is the HL signature — shown below.
Why HL works when SSA fails
With a hypotenuse and one leg, the third side is not actually unknown. The Pythagorean Theorem forces it: if the hypotenuse is and one leg is , the other leg must be . Both triangles are forced to have the same three side lengths.
So HL is really SSS wearing a disguise — the right angle plus two known sides pins down the third side, and three pairs of congruent sides always prove congruence. Ordinary SSA has no such guarantee, which is why it fails in general.
HL versus SAS in right triangles
If you are given both legs of each right triangle instead of a hypotenuse, do not reach for HL. The two legs meet at the right angle, and all right angles are congruent — so two congruent legs with the congruent right angle between them is exactly SAS.
Quick sorting rule: hypotenuse and a leg means HL; leg and a leg means SAS. Either way the triangles are congruent, but the named reason must match the given parts.
Worked examples
Example 1: recognizing HL
Two right triangles each have a hypotenuse of in and a leg of in. Which criterion proves them congruent?
Answer: HL
Example 2: HL is secretly SSS
A right triangle has hypotenuse and one leg . Show the third side is forced.
Answer: The remaining leg must be , so all three sides are determined.
Example 3: two legs is SAS, not HL
In two right triangles, both pairs of corresponding legs are congruent, but nothing is known about the hypotenuses. Which criterion applies?
Answer: SAS — HL is not needed because no hypotenuse is given.
Try one yourself
Common questions
Does HL work on triangles that are not right triangles?
No. Without the right angle, hypotenuse-leg information is just SSA — two sides and a non-included angle — which can produce two different triangles. The right angle is what makes the third side forced.
Which side is the hypotenuse?
The side opposite the right angle. It is always the longest side of a right triangle, and it never touches the right angle's vertex.
Do both pairs of legs need to be congruent for HL?
No — one pair of corresponding legs is enough, as long as the hypotenuses are also congruent and both triangles are right triangles. If both pairs of legs are congruent, you would cite SAS instead.
Is HL an extra theorem or a shortcut of another one?
It is proved from the Pythagorean Theorem: hypotenuse plus one leg determines the third side, so HL reduces to SSS. It gets its own name because the setup is so common in right-triangle problems and proofs.
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