Question

directions: use the figure and information to set create a formal proof.
prove the triangles are congruent, given \\( \overline { p q } \parallel \overline { n m } \\) and \\( \overline { p q } \cong \overline { n m } \\)

Explanation:

Step1: Identify alternate interior angles

Since \( \overline{PQ}\parallel\overline{NM}\), by the Alternate Interior Angles Theorem, \( \angle M\cong\angle P\) (when the transversal is \( \overline{PM}\)) and \( \angle N\cong\angle Q\) (when the transversal is \( \overline{NQ}\)).

Step2: Use ASA congruence criterion

We know \( PQ\cong NM\) (given). In \( \triangle MNO\) and \( \triangle PQO\), we have two pairs of congruent angles (\( \angle M\cong\angle P\), \( \angle N\cong\angle Q\)) and a pair of congruent included sides (\( PQ\cong NM\)). By the Angle - Side - Angle (ASA) congruence criterion, \( \triangle MNO\cong\triangle PQO\).

Answer:

	Statement	Reason
2	\( PQ\cong NM\)	Given
3	\( \angle N\cong\angle Q\)	Alternate Interior Angles Theorem (because \( \overline{PQ}\parallel\overline{NM}\) and transversal \( \overline{NQ}\))
4	\( \triangle MNO\cong\triangle PQO\)	ASA (Angle - Side - Angle) congruence criterion