Lesson on Hypotenuse Leg or HL congruence theorem

In this write-up, we’ll discover the hypotenuse leg (HL) theory. Like, SAS, SSS, ASA, and AAS, it is likewise among the congruency postulates of a triangle. Let’s learn more about the HL congruence theorem.

The difference is that the various other four proposes relate to all triangular. All at once, the Hypotenuse Leg Theory holds for the best triangular only because, undoubtedly, the hypotenuse is just one of the right-angled triangular legs.

What is the HL congruence theorem?

The hypotenuse leg thesis is a standard utilized to confirm whether an offered set of right triangular is coinciding.

The hypotenuse leg (HL) theory states that; an offered collection of triangles conforms if their hypotenuse and one leg’s equivalent lengths are equal.

Unlike other congruency postulates such as; SSS, SAS, ASA, and AAS, three quantities are examined, with hypotenuse leg (HL) theory, two sides of a right triangular are just considered.

Proof of Hypotenuse Leg Theory

In the diagram over, triangular ABC and QPR are right triangular with AD = RQ, AC = PQ.

By Pythagorean Thesis,

AC2 = AB2 + BC2 and PQ2 = RQ2 + RP2

Considering That AC = PQ, an alternative to obtaining.

AB2 + BC2 = RQ2 + RP2

But, AD = RQ,

By alternative.

RQ2 + BC2 = RQ2 + RP2

Accumulate like terms to get;

BC2 =RP2.

Thus, △ ABC ≅ △ QPR.

Example 1.

If PR⊥ QS, verify that QPR, as well as PR, are conforming.


Triangle QPR and PRS are right triangular because they both have a 90-degree angle at point R.


PQ = PS (Hypotenuse).

PR = PR (Common side).

Therefore, by Hypotenuse– Leg (HL) theorem, △ QPR ≅ △ PR.

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