PQRS is a parallelogram , if `anglePQS = 40^(@) and PS = QS` , then `angleSQR -angle QRS` = ___________.

Options

A. `60^(@)`
B. `20^(@)`
C. `30^(@)`
D. `45^(@)`

Answer

To solve the problem, we need to find the value of \( \angle SQR - \angle QRS \) in the parallelogram PQRS, given that \( \angle PQS = 40^\circ \) and \( PS = QS \).

### Step-by-Step Solution:

1. **Identify the Properties of the Parallelogram**:
- In a parallelogram, opposite sides are equal and opposite angles are equal.
- Given that \( PS = QS \), we can conclude that triangle PQS is isosceles.

2. **Determine Angles in Triangle PQS**:
- Since \( PS = QS \), the angles opposite these sides are equal. Thus, \( \angle PQS = \angle PSQ \).
- Given \( \angle PQS = 40^\circ \), we can denote \( \angle PSQ = 40^\circ \).

3. **Calculate the Remaining Angle in Triangle PQS**:
- The sum of angles in a triangle is \( 180^\circ \).
- Therefore, we can find \( \angle QPS \):
\[ \angle QPS = 180^\circ - \angle PQS - \angle PSQ = 180^\circ - 40^\circ - 40^\circ = 100^\circ. \]

4. **Identify Angles in the Parallelogram**:
- Since \( PQRS \) is a parallelogram, opposite angles are equal. Thus, \( \angle QRS = \angle PQR = 40^\circ \) and \( \angle SQR = \angle QPS = 100^\circ \).

5. **Calculate \( \angle SQR - \angle QRS \)**:
- Now we can find the difference:
\[ \angle SQR - \angle QRS = 100^\circ - 40^\circ = 60^\circ. \]

### Final Answer:
\[ \angle SQR - \angle QRS = 60^\circ. \]