HL Theorem (Hypotenuse-Leg)
In short: The HL theorem is a congruence test for right triangles only: matching hypotenuses plus one matching pair of legs make the two triangles identical.
GeometryGeometryG-SRT.B.5
What is HL Theorem?
The HL theorem, short for hypotenuse-leg, is a shortcut reserved for right triangles. If the hypotenuse and one leg of a right triangle equal the hypotenuse and one leg of another right triangle, the two triangles are congruent.
It earns its own name because side-side-angle fails in general. Here the known angle is the right angle, which faces the longest side, and the Pythagorean relation \(a^2 + b^2 = c^2\) then forces the unknown leg to match as well.
How to spot it
- Two right triangles are drawn, each with a small square at one corner.
- The givens list one pair of equal hypotenuses and one pair of equal legs, with no angle pair besides the right angles.
- Ladders, ramps or braces of equal length lean against upright walls on flat ground.
- A proof needs a reason, and SAS, ASA and AAS do not fit the marked parts.
The method
- Confirm that each triangle holds a right angle, from a square mark or perpendicular givens; HL needs it.
- Identify each hypotenuse, the side facing the right angle, and show the two are equal.
- Show that one leg of the first triangle equals one leg of the second.
- State the congruence in matching vertex order and cite HL as the reason.
HL Theorem: solved example
Problem
Two 17-foot ladders lean against two different vertical walls on level ground. Each ladder touches its wall 15 feet above the ground. Show the two triangles formed are congruent and find how far each ladder base sits from its wall.
Solution
Each wall and the ground meet at a right angle, so both triangles are right triangles. Hypotenuses: \(17 = 17\). Legs: \(15 = 15\). By HL the triangles are congruent.
The missing leg follows from the Pythagorean theorem: \(\sqrt{17^2 – 15^2} = \sqrt{289 – 225} = \sqrt{64} = 8\). Both ladder bases are 8 feet from the wall.
Try it
Question 1
Two right triangles each have a hypotenuse of 10 cm and a leg of 6 cm. What is the other leg of each, and are the triangles congruent?
Show the answer
\(\sqrt{100 – 36} = 8\) cm for both. They are congruent by HL.
Question 2
Two right triangles both have legs of 9 and 12. Is HL the best reason for their congruence?
Show the answer
No. The right angle lies between the two legs, so SAS applies directly, and no hypotenuse is needed.
HL Theorem vs. Triangle Congruence
The four general tests, SSS, SAS, ASA and AAS, work for every triangle. Side-side-angle is banned there, since a lone angle outside the two sides can fit two different shapes. The HL theorem is the safe special case: a right angle facing the longest side leaves room for only one shape.
Keep going
Lessons
Worksheets and quizzes
- Congruent Triangles and Rigid Motions – Worksheet
- Congruent Triangles and Rigid Motions – Quiz
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry