What additional information do you need to prove that ∆lmo ≅ ∆lno by the hl theorem? ∠nlo ≅ ∠mlo ∠mol ≅ ∠nol

The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.
The triangles ΔLNO and ΔLMO have the same leg Lo, therefore you need the equality of hypotenuses LM=LN.

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