Question

directions: use the figure and information to set create a formal proof.
prove the triangles are congruent, given ( overline{pq} parallel overline{nm} ) and ( overline{pq} cong overline{nm} )

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 \( \overline{PM}\)).

Step2: Identify another pair of alternate interior angles

Also, since \( \overline{PQ}\parallel\overline{NM}\), by the Alternate Interior Angles Theorem, \( \angle N\cong\angle Q\) (when the transversal is \( \overline{NQ}\)).

Step3: Use ASA (Angle - Side - Angle) congruence criterion

We know \( PQ\cong NM\) (given). In \( \triangle MNO\) and \( \triangle PQO\), we have \( \angle M\cong\angle P\), \( PQ\cong NM\), \( \angle N\cong\angle Q\). So, by ASA (Angle - Side - Angle) congruence, \( \triangle MNO\cong\triangle PQO\).

Answer:

1. \( \angle M\cong\angle P\) (Reason: Alternate Interior Angles Theorem)
2. \( \angle N\cong\angle Q\) (Reason: Alternate Interior Angles Theorem)
3. \( \triangle MNO\cong\triangle PQO\) (Reason: ASA (Angle - Side - Angle) congruence)