HL (Hypotenuse-Leg) Congruence
For right triangles only, the hypotenuse and one leg are enough to prove congruence — even though the known angle (90°) is not between the known sides.
Introduction
We said SSA doesn't work in general — it's ambiguous. But for right triangles, the 90° angle eliminates the ambiguity. The HL Theoremis a special case that works exclusively for right triangles. It's like a “fixed” version of SSA.
Past Knowledge
SSS, SAS, ASA, AAS (5.2.2–5.2.5). Right triangles. Pythagorean Theorem preview.
Today's Goal
Prove and apply the HL Congruence Theorem for right triangles.
Future Success
CPCTC proofs (5.2.7), Pythagorean Theorem (8.1), and trigonometry.
Key Concepts
HL Congruence Theorem
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Requirements: Both triangles must be right triangles.
Vocabulary Reminder
- Hypotenuse — the side opposite the right angle (always the longest side)
- Legs — the two sides that form the right angle
Why HL Works (But General SSA Does Not)
In SSA, two triangles can have the same two sides and non-included angle. But the right angle forces the third side via the Pythagorean Theorem: . If you know the hypotenuse and a leg , then is forced — giving you SSS.
Theorem & Proof
Two-Column Proof: HL Congruence Theorem
Given: Right (right angle at ) and right (right angle at ) with (hypotenuses) and (legs)
Prove:
The right angle + Pythagorean Theorem forces the third side, reducing HL to SSS.
| # | Statement | Reason |
|---|---|---|
| 1 | , , | Given |
| 2 | and | Pythagorean Theorem (in both right triangles) |
| 3 | , so | Substitution ( and ); lengths are positive, so equal squares imply equal lengths |
| 4 | Steps 1 and 3 (all three pairs of sides are congruent) | |
| 5 | SSS Congruence (step 4) |
∎ HL reduces to SSS via the Pythagorean Theorem. It only works for right triangles because that's when the Pythagorean relationship holds.
Worked Examples
Direct HL Application
Right and right both have right angles at and . and . Prove congruence.
Both are right triangles. ✓
Hypotenuses: ✓
Legs: ✓
by HL. (The other leg: , via )
HL with Altitude
In isosceles with , altitude is drawn to base . Prove .
is an altitude, so . Both sub-triangles are right triangles.
Hypotenuse: (given — legs of isosceles triangle)
Leg: (Reflexive Property)
by HL.
Why SSA Fails (But HL Doesn't)
Two non-right triangles have , , . Are they congruent?
This is SSA (the angle is NOT included between the two sides, and it's NOT 90°).
There could be two different triangles with these measurements (the “ambiguous case”).
Without a right angle, we cannot use the Pythagorean Theorem to force the third side.
Cannot prove congruence. SSA is ambiguous for non-right triangles. HL only works when the angle is 90°.
Common Pitfalls
Using HL on Non-Right Triangles
HL only works for right triangles. If neither triangle has a confirmed 90° angle, you cannot use HL. Check for right angle marks or perpendicular statements first.
Confusing Hypotenuse and Leg
The hypotenuse is always the side opposite the right angle (the longest side). The two legs are the sides that form the right angle. You need one of each for HL — not two legs.
Real-Life Applications
Construction — Staircase Stringers
When building stairs, each step forms a right triangle. If two stringers have the same rise (leg) and are cut from an equally angled board (hypotenuse), HL guarantees the steps are congruent — essential for a level staircase.
Wheelchair Ramp Design
ADA ramp specifications define a maximum slope (rise:run ratio). If two ramps have the same total length (hypotenuse) and the same vertical rise (leg), HL ensures they have identical slope angles — guaranteeing equal accessibility.
Practice Quiz
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