To use the ASA (Angle-Side-Angle) congruence theorem to prove that triangles LKN and PQM are congruent, we need two angles and the included side between them to be congruent. You've mentioned that angles LKN (which is given as a right angle) and PQM (also a right angle) are already congruent (both are 90 degrees). So we have one pair of congruent angles.
We also know that sides KL and QP are congruent, which means we should consider what specifically "NL" and "MP" or "NK" and "MQ" represent in relation to the angles in terms of their positions.
To fully apply the ASA theorem, we need one of the following pieces of information:
- Show that NL is congruent to MP.
- Show that NK is congruent to MQ.
- Provide that angle N is congruent to angle M.
- Provide that angle L is congruent to angle P.
In conclusion, to definitively use the ASA theorem, you need to establish that one additional angle (either N ≅ M or L ≅ P) and the side opposite to one of these angles (be it NL ≅ MP or NK ≅ MQ) are congruent. This information will ensure that the two triangles LKN and PQM can be proven congruent using ASA.
Thus, the additional information needed is either Angle N ≅ Angle M or Angle L ≅ Angle P and the measurement comparing either NL ≅ MP or NK ≅ MQ.
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