Question

1. given: \\( \overline{mo} \\) is the perpendicular bisector of \\( \overline{np} \\) prove: \\( \overline{mn} \cong \overline{mp} \\) \\( \overline{mo} \\) is perpendicular bisector of \\( \overline{np} \\) given

Explanation:

Step1: Use the definition of perpendicular bisector

Since \(\overline{MO}\) is the perpendicular bisector of \(\overline{NP}\), we have \(\angle 3=\angle 4 = 90^{\circ}\) and \(NO = OP\).

Step2: Consider triangles \(\triangle MNO\) and \(\triangle MPO\)

In \(\triangle MNO\) and \(\triangle MPO\):

• \(NO=OP\) (from the definition of perpendicular bisector)
• \(\angle 3=\angle 4\) (from Step 1)
• \(MO = MO\) (common side)

Step3: Apply the Side - Angle - Side (SAS) congruence criterion

By the SAS congruence criterion, \(\triangle MNO\cong\triangle MPO\).

Step4: Use the property of congruent triangles

If two triangles are congruent, then their corresponding sides are equal. So, \(\overline{MN}\cong\overline{MP}\) (corresponding parts of congruent triangles \(\triangle MNO\) and \(\triangle MPO\)).

Answer:

\(\overline{MN}\cong\overline{MP}\) is proved as above.