PRINCIPIA · THEOREM

HL (hypotenuse-leg) congruence (right triangles)

Dependencies: Pythagorean theorem, SSS congruence.

Statement

In right triangles Rt△ABC\mathrm{Rt}\triangle ABC and Rt△A′B′C′\mathrm{Rt}\triangle A'B'C', suppose

∠C=∠C′=90∘,∣AB∣=∣A′B′∣=c,∣BC∣=∣B′C′∣=a.\angle C = \angle C' = 90^\circ,\qquad |AB| = |A'B'| = c,\qquad |BC| = |B'C'| = a.

That is, the two right triangles have equal corresponding hypotenuses and one pair of equal corresponding legs. Then

△ABC  ≅  △A′B′C′.\triangle ABC \;\cong\; \triangle A'B'C'.

Notation: "HL" stands for Hypotenuse-Leg. Alongside the four general-triangle congruence criteria SSS / SAS / ASA / AAS, the HL congruence criterion is a fifth shortcut, available only for right triangles — locking in congruence with just two sides.

HL congruence: in two Rt△s, equal hypotenuse c + one equal leg a ⇒ third side b also equal ⇒ congruent.

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