Key takeaway: The HL theorem (hypotenuse-leg) says two right triangles are congruent when their hypotenuses are equal and one leg of each triangle is equal, so two matching parts are enough.
Grade 10 – GeometryG-SRT.B.5Geometry and Measurement
Definition of HL Theorem
The HL theorem, short for hypotenuse-leg, is a congruence shortcut reserved for right triangles. If the longest sides match and one pair of legs match, the two triangles are copies of each other.
It needs only two facts because the right angle is already guaranteed in both triangles, and the Pythagorean relationship then forces the other leg to match. Both triangles must truly contain a right angle, and the equal leg must be matched with a leg, not with the hypotenuse.
Formula and notation
For right triangles \(\triangle PQR\) (right angle at \(Q\)) and \(\triangle STU\) (right angle at \(T\)):
\[ PR = SU \ \text{ and } \ PQ = ST \ \Rightarrow \ \triangle PQR \cong \triangle STU \]
In a proof the reason is written “HL” or “Hypotenuse-Leg Theorem.”
Key facts
- The HL theorem works only for right triangles; the right angle must be stated or marked.
- The hypotenuse is the side opposite the right angle and the longest side.
- It is the one valid pattern of the shape side-side-angle in right triangles.
- After HL, the third sides and the other angles are also equal.
HL Theorem example
Use the HL theorem on this pair. Right triangle \(PQR\) has its right angle at \(Q\), with \(PR = 17\) ft and \(PQ = 8\) ft. Right triangle \(STU\) has its right angle at \(T\), with \(SU = 17\) ft and \(ST = 8\) ft. Show the triangles are congruent and find \(TU\).
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Both triangles are right triangles, \(PR = SU\) (hypotenuses) and \(PQ = ST\) (legs). So \(\triangle PQR \cong \triangle STU\) by HL.
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In \(\triangle PQR\): \(QR^2 = 17^2 – 8^2 = 289 – 64 = 225\), so \(QR = 15\) ft.
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Corresponding sides of congruent triangles are equal, so \(TU = QR\).
The triangles are congruent by HL, and \(TU = 15\) ft.
Common mistake
Watch out
Rhea sees two triangles with equal hypotenuses of 17 ft and concludes they are congruent by HL, although one has legs 8 ft and 15 ft and the other has legs 10 ft and about 13.7 ft.
Why it is wrong
HL requires a matching leg as well as the hypotenuse. A single equal side is not enough. The leg lengths here differ, so the triangles are not congruent, even though both are right triangles with hypotenuse 17 ft.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement