Question

given: ( \triangle nop cong \triangle qrp );
n is the midpoint
of ( overline{mo} ).
prove: mnqr is a
parallelogram.

Explanation:

Step1: Use CPCTC

Since \(\triangle NOP\cong\triangle QRP\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(MN = QR\).

Step2: Use Mid - point Property

Since \(N\) is the mid - point of \(\overline{MO}\), \(PN = FN\) (from the congruent triangles \(\triangle NOP\cong\triangle QRP\), \(OP = RP\) and \(N\) divides \(MO\) such that \(MN = NO\), and from the congruence \(QR=NO\)). Also, \(MQ\parallel NR\) (alternate interior angles are equal from the congruent triangles).

Step3: Use Parallelogram Definition

A quadrilateral \(MNQR\) with \(MN = QR\) and \(MQ\parallel NR\) satisfies the definition of a parallelogram (a quadrilateral with one pair of opposite sides equal and parallel is a parallelogram).

Answer:

1. \(MN = QR\)
2. \(MQ\parallel NR\); Alternate interior angles from \(\triangle NOP\cong\triangle QRP\)
3. A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.