Answer:
\(\overline{LQ}\parallel \overline{NP}\), given
∠LMQ ≅ ∠PMN, vertical angles
∠Q ≅ ∠N, alternate interior angles
ΔLMQ ~ ΔPMN, AA~
Step-by-step explanation:
We know that we need to prove ΔLMQ ~ ΔPMN, in words, the triangle LMQ is similar to triangle PMN.
It is given that \(\overline{LQ}\parallel \overline{NP}\), so we will put given as the first justification with the statement \(\overline{LQ}\parallel \overline{NP}\).
The second thing I notice is the vertical angles. We can prove the statement ∠LMQ ≅ ∠PMN with the justification of vertical angles. Vertical angles will always be congruent. Vertical angles are made by two intersecting lines.
Since \(\overline{LQ}\parallel \overline{NP}\) are two parallel lines, line QN is a transversal. This means that it forms alternate interior angles. With this information, ∠Q ≅ ∠N.
Finally, we come to proving ΔLMQ ≅ ΔPMN. We have proved that each of the two triangles have two congruent angles, this means they are proved similar by the AA theorem, aka angle-angle. When two triangles have two congruent angles, the third must also be congruent meaning the triangles are similar.
