Question

1. in the figure, what additional piece would need to be congruent in order for \\( \triangle l p n \cong \triangle l m n \\) using the asa congruence theorem. \\( \overline{l n} \cong \overline{l n} \\) \\( \overline{l p} \cong \overline{l m} \\) \\( \overline{n p} \cong \overline{n m} \\)

Explanation:

Step1: Recall ASA Congruence Theorem

ASA (Angle - Side - Angle) congruence theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Step2: Analyze the given triangles \( \triangle LPN\) and \( \triangle LMN\)

We already have one pair of congruent angles (marked in the figure) and the side \( LN\) is common to both triangles (\( LN\cong LN\) by the reflexive property of congruence). For ASA, we need the sides between the two angles to be congruent.

Step3: Check the options

• \( LN\cong LN\) is the reflexive property but not the additional side needed for ASA.
• \( \overline{LP}\cong\overline{LM}\) is the side between the two angles (the marked angles and the vertical - angle at \( L\)) in \( \triangle LPN\) and \( \triangle LMN\) respectively.
• \( \overline{NP}\cong\overline{NM}\) is not the included side for the ASA criterion.

Answer:

\(\overline{LP}\cong\overline{LM}\)