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HL Congruence

HL Congruence

Hypotenuse-leg congruence. When two right triangles have corresponding sides that are congruent as shown below, the triangles are congruent.

 

Two right triangles ABC and DEF with tick marks showing congruent hypotenuses and legs; △ABC ≅ △DEF

 

 

See also

Congruence tests for triangles

Key Formula

If △ABC and △DEF are right triangles with ∠C=∠F=90°, and AB=DE (hypotenuse), and BC=EF (leg), then △ABC≅△DEF.\text{If } \triangle ABC \text{ and } \triangle DEF \text{ are right triangles with } \angle C = \angle F = 90°, \text{ and } AB = DE \text{ (hypotenuse)}, \text{ and } BC = EF \text{ (leg)}, \text{ then } \triangle ABC \cong \triangle DEF.
Where:
  • AB,DEAB, DE = The hypotenuses of the two right triangles
  • BC,EFBC, EF = A pair of corresponding legs of the two right triangles
  • ∠C,∠F\angle C, \angle F = The right angles (each equal to 90°) in the respective triangles
  • ≅\cong = Symbol meaning 'is congruent to' — same shape and size

Worked Example

Problem: Right triangle ABC has a right angle at C, with hypotenuse AB = 13 and leg BC = 5. Right triangle DEF has a right angle at F, with hypotenuse DE = 13 and leg EF = 5. Prove the two triangles are congruent.
Step 1: Verify both triangles are right triangles. Triangle ABC has a right angle at C, and triangle DEF has a right angle at F.
∠C=90°,∠F=90°\angle C = 90°, \quad \angle F = 90°
Step 2: Compare the hypotenuses. Both hypotenuses are equal in length.
AB=DE=13AB = DE = 13
Step 3: Compare one pair of corresponding legs. Both legs are equal in length.
BC=EF=5BC = EF = 5
Step 4: Apply the HL Congruence Theorem. Since both triangles are right triangles with congruent hypotenuses and a congruent pair of legs, the triangles are congruent.
△ABC≅△DEF(by HL)\triangle ABC \cong \triangle DEF \quad \text{(by HL)}
Answer: By the HL Congruence Theorem, △ABC ≅ △DEF. (Note: the other leg in each triangle must be 12, by the Pythagorean theorem.)

Another Example

This example shows how HL is used when two triangles share a common hypotenuse — a very common setup in geometry proofs involving overlapping or adjacent triangles.

Problem: In the figure, two right triangles share a common hypotenuse. Triangle PQR has a right angle at R, and triangle PSR has a right angle at S. Both triangles share hypotenuse PR, and QR = SR = 8. Are the triangles congruent?
Step 1: Confirm both triangles are right triangles. Triangle PQR has a right angle at R (wait — let's re-read). Actually, triangle PQR has a right angle at Q, and triangle PSR has a right angle at S. Let me set this up correctly: right angle at Q means PQ is a leg and QR is a leg, with PR as the hypotenuse. Right angle at S means PS is a leg and SR is a leg, with PR as the hypotenuse.
∠Q=90°,∠S=90°\angle Q = 90°, \quad \angle S = 90°
Step 2: Identify the shared hypotenuse. Both triangles have PR as their hypotenuse, so the hypotenuses are automatically congruent.
PR=PR(common side)PR = PR \quad \text{(common side)}
Step 3: Compare a pair of corresponding legs.
QR=SR=8QR = SR = 8
Step 4: Apply the HL Congruence Theorem. Both are right triangles sharing a hypotenuse, with one pair of congruent legs.
△PQR≅△PSR(by HL)\triangle PQR \cong \triangle PSR \quad \text{(by HL)}
Answer: Yes, △PQR ≅ △PSR by the HL Congruence Theorem.

Frequently Asked Questions

Why does HL Congruence work with only two sides and no included angle?
In a right triangle, the right angle is already known, and the Pythagorean theorem guarantees that if the hypotenuse and one leg match, the other leg must also match. So you effectively know all three sides. This is why HL is really a special case of SSS (Side-Side-Side) congruence that only applies to right triangles.
Can you use HL Congruence on triangles that are not right triangles?
No. The HL theorem requires both triangles to have a 90° angle. Without the right angle, knowing just two sides (a longest side and one other side) is not enough to prove congruence — you would need SSA, which is not a valid congruence test in general.
What is the difference between HL Congruence and SAS Congruence?
SAS (Side-Angle-Side) requires two sides and the included angle between them to be congruent. HL requires the hypotenuse and a leg, where the known angle (90°) is not between the two given sides — it is opposite the hypotenuse. SAS works for any triangle, while HL works only for right triangles.

HL Congruence vs. SAS Congruence

HL CongruenceSAS Congruence
What you needHypotenuse + one leg (and both right angles)Two sides + the included angle between them
Triangle typeRight triangles onlyAny triangle
Angle positionThe known angle (90°) is opposite the hypotenuse, not between the two known sidesThe known angle is between the two known sides
Why it worksThe Pythagorean theorem forces the third side to be determinedThe included angle locks the shape of the triangle

Why It Matters

HL Congruence appears frequently in geometry courses whenever right triangles are involved — for instance, when proving properties of isosceles triangles by dropping an altitude, or when working with perpendicular bisectors. It is one of the five standard triangle congruence theorems (SSS, SAS, ASA, AAS, HL), and it is the only one that applies exclusively to right triangles. Many standardized tests and proof-based problems expect you to recognize when HL is the most efficient path to proving congruence.

Common Mistakes

Mistake: Using HL on triangles that are not right triangles.
Correction: HL requires both triangles to contain a 90° angle. Always verify the right angle exists before applying this theorem. If there is no right angle, use SSS, SAS, ASA, or AAS instead.
Mistake: Confusing HL with SSA (Side-Side-Angle), which is generally not a valid congruence test.
Correction: HL looks like SSA because the right angle is not between the two known sides. However, HL works specifically because the 90° angle plus the Pythagorean theorem eliminates the ambiguity that makes SSA invalid in general. Only apply this reasoning when the angle is exactly 90°.

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