Question

given: ( \triangle nopcong\triangle qrp ); n is the midpoint of ( overline{mo} ). prove: mnqr is a parallelogram. 1. ( \triangle nopcong\triangle qrp ) 1. given 2. click here to insert 2. corresponding parts of congruent tri - angles are congruent (c.p.c.t.c.) 3. n is the midpoint of ( overline{mo} ). 3. given 4. click here to insert 4. click here to insert 5. click here to insert 5. transitive property

Explanation:

Step1: Use CPCTC

Since $\triangle NOP\cong\triangle QRP$, by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we have $NO = QR$ and $\angle NOP=\angle QRP$.

Step2: Use mid - point property

Because $N$ is the mid - point of $\overline{MO}$, we know that $MN = NO$.

Step3: Substitute

From Step1 ($NO = QR$) and Step2 ($MN = NO$), by substitution, we get $MN = QR$.

Step4: Use angle - side relationship

Since $\angle NOP=\angle QRP$, we can say that $MN\parallel QR$ (alternate interior angles are equal for parallel lines).

Step5: Apply parallelogram criterion

A quadrilateral with one pair of opposite sides ($MN$ and $QR$) both equal and parallel is a parallelogram.

Answer:

1. $NO = QR,\angle NOP=\angle QRP$; 4. $MN = NO$ (Reason: Definition of mid - point); 5. $MN = QR$ and $MN\parallel QR$ (so $MNQR$ is a parallelogram)