In a parallelograms PQRS, M and N are points on PQ and RS such that PM = RN. Prove that MS || NQ.

Answer

To prove that \( MS \parallel NQ \) in the parallelogram \( PQRS \) where \( PM = RN \), we will follow these steps:

### Step 1: Understand the Given Information
We have a parallelogram \( PQRS \) with points \( M \) on \( PQ \) and \( N \) on \( RS \) such that \( PM = RN \).

**Hint:** Identify the properties of a parallelogram, particularly that opposite sides are equal and parallel.

### Step 2: Assign Lengths
Let \( PM = RN = x \). Since \( PQRS \) is a parallelogram, the lengths of opposite sides are equal. Let \( PQ = RS = L \).

**Hint:** Use variables to represent lengths to simplify calculations.

### Step 3: Calculate Remaining Lengths
Now, we can find the lengths of \( MQ \) and \( NS \):
- \( MQ = PQ - PM = L - x \)
- \( NS = RS - RN = L - x \)

**Hint:** Use the property of segments to express the remaining lengths in terms of the total length and the known segment.

### Step 4: Establish Equality of Segments
From the calculations, we have:
- \( MQ = L - x \)
- \( NS = L - x \)

Thus, \( MQ = NS \).

**Hint:** Note that if two segments are equal, they can be used to show parallelism in the next steps.

### Step 5: Use the Parallelogram Properties
In the quadrilateral \( MQNS \):
- We have \( MQ = NS \) (opposite sides are equal).
- We know \( PQ \parallel RS \) (as they are opposite sides of the parallelogram).

**Hint:** Recall the condition for a quadrilateral to be a parallelogram: if one pair of opposite sides is both equal and parallel, then the quadrilateral is a parallelogram.

### Step 6: Conclude that \( MQNS \) is a Parallelogram
Since \( MQ = NS \) and \( PQ \parallel RS \), by the properties of parallelograms, \( MQNS \) is a parallelogram.

**Hint:** Use the definition of a parallelogram to conclude that if one pair of opposite sides is both equal and parallel, the shape is a parallelogram.

### Step 7: Prove \( MS \parallel NQ \)
In parallelogram \( MQNS \), the opposite sides \( MS \) and \( NQ \) are parallel by definition.

**Hint:** Remember that in any parallelogram, opposite sides are always parallel.

### Conclusion
Thus, we have proved that \( MS \parallel NQ \).