HL means ‘Hypotenuse-Leg’. HL triangles are right triangles where hypotenuse and one of the other two legs are known. Shown below is a HL triangle, △ABC, where AC is the hypotenuse and AB is one of the other two legs.

HL Triangle Congruence Theorem

Unlike other congruency postulates, where three quantities are tested, in HL theorem two sides of a right triangle are considered.

Proving Triangles Congruent using the Hypotenuse-Leg Theorem

Prove HL Triangle Congruence Theorem

To prove:

ΔABC ≅ ΔEGF

Proof:

Given: ΔABC and ΔEGF are right triangles where AB ≅ EG, AC ≅ EF By Pythagorean Theorem, In ΔABC, AC^{2 }= AB^{2} + BC^{2 }……. (1), and In ΔPQR, EF^{2 }= EG^{2} + GF^{2 }……. (2) Since, AC ≅ EF, from (1) and (2) we can write, AB^{2} + BC^{2} = EG^{2} + GF^{2 }……. (3) Again, AB ≅ EG Substituting in (3) we get, EG^{2} + BC^{2} = EG^{2} + GF^{2} BC^{2} = GF^{2} Thus, ΔABC ≅ ΔEGF Hence Proved

Let us take some examples to understand the concept.

Solved Examples and Practice Proof

Prove whether ∆PQR and ∆YXZare congruent.

Solution:

Given that ∆PQR and ∆YXZ are right triangles, where PR ≅ YZ (equal hypotenuse) PQ ≅ YX (equal leg) Thus, by ‘Hypotenuse-Leg’ theorem, ∆PQR ≅ ∆YXZ Hence proved

Prove that in the given figure, ∆EGF ≅ ∆EHF. Given, EF ⊥ GH and EG ≅ EH.

Solution:

Given that EF ⊥ GH and EG ≅ EH Thus, ∆EGF and ∆EHF are right triangles as ∠EFG = ∠EFH = 90° EG ≅ EH (equal hypotenuse) EF ≅ EF (common leg)

Find the value of x in the given pairs of congruent triangles.

Solution:

Given that ∆ABC ≅ ∆NLM Thus, x + 4 = 3x – 16 => x – 3x = – 16 – 4 => -2x = -20 => x = 10 Thus, the value of x = 10

Show that ∆OXW and ∆OYZ are congruent. Given that ∠X = ∠Y = 90° and O is the midpoint of XY and WZ.

Solution:

Given that ∆OXW and ∆OYZ are right triangles ∠X = ∠Y = 90° XO ≅ YO (equal leg) WO ≅ ZO (equal hypotenuse) Thus, by the ‘Hypotenuse-Leg’ theorem, ∆OXW ≅ ∆OYZ Hence proved