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 \(\overline{PM}\)) and \(\angle N\cong\angle Q\) (when the transversal is \(\overline{NQ}\)).

Step2: Use the ASA (Angle - Side - Angle) congruence criterion

We know that \(\overline{PQ}\cong\overline{NM}\) (given). For \(\triangle MNO\) and \(\triangle PQO\), we have \(\angle M\cong\angle P\), \(\overline{PQ}\cong\overline{NM}\), and \(\angle N\cong\angle Q\). So, by the ASA (Angle - Side - Angle) congruence theorem, \(\triangle MNO\cong\triangle PQO\).

Answer:

• \(\angle M\cong\angle P\) (Reason: Alternate - Interior Angles Theorem)
• \(\angle N\cong\angle Q\) (Reason: Alternate - Interior Angles Theorem)
• \(\triangle MNO\cong\triangle PQO\) (Reason: ASA (Angle - Side - Angle) congruence theorem)