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 } \\) complete proof must be shown on your work paper to recieve credit for this assignment you have 3 attempts to correctly answer this problem.

Explanation:

Step1: Alternate interior angles

Since \(PQ\parallel NM\), by the alternate - interior angles theorem, \(\angle M\cong\angle P\) (when the transversal is \(PM\)).

Step2: Alternate interior angles

Also, since \(PQ\parallel NM\), by the alternate - interior angles theorem, \(\angle N\cong\angle Q\) (when the transversal is \(NQ\)).

Step3: ASA congruence

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

Answer:

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