PQRS is a rhombus in which `/_SPQ = 64^@`. Equilateral triangles PXQ and QYR are drawn outside the rhombus on sides PQ and QR. Calculate the magnitude of the angle `/_QXY`.

Options

A. `26^@`
B. `28^@`
C. `30^@`
D. `45^@`

Answer

To solve the problem, we need to find the magnitude of the angle \( \angle QXY \) in the given rhombus \( PQRS \) with the angle \( \angle SPQ = 64^\circ \) and equilateral triangles \( PXQ \) and \( QYR \) drawn outside the rhombus.

### Step-by-Step Solution:

1. **Understanding the Rhombus and Angles**:
- In rhombus \( PQRS \), all sides are equal, and opposite angles are equal. Given \( \angle SPQ = 64^\circ \), we know that \( \angle PQR = \angle SPQ = 64^\circ \).
- The adjacent angles in a rhombus are supplementary, so:
\[ \angle PQR + \angle QRS = 180^\circ \]
Therefore:
\[ \angle QRS = 180^\circ - 64^\circ = 116^\circ \]

2. **Angles in Equilateral Triangles**:
- Since \( PXQ \) and \( QYR \) are equilateral triangles, each angle in these triangles is \( 60^\circ \).
- Thus, we have:
\[ \angle QXP = 60^\circ \quad \text{and} \quad \angle QYR = 60^\circ \]

3. **Finding Angle \( \angle XQY \)**:
- To find \( \angle XQY \), we can use the fact that the angles around point \( Q \) must sum to \( 360^\circ \):
\[ \angle SPQ + \angle QXP + \angle XQY + \angle QYR = 360^\circ \]
- Plugging in the known values:
\[ 64^\circ + 60^\circ + \angle XQY + 60^\circ = 360^\circ \]
- Simplifying this:
\[ 184^\circ + \angle XQY = 360^\circ \]
- Thus:
\[ \angle XQY = 360^\circ - 184^\circ = 176^\circ \]

4. **Finding Angles \( \angle X \) and \( \angle Y \)**:
- Since \( \angle XQY \) is \( 176^\circ \) and angles \( \angle X \) and \( \angle Y \) are equal (as they are opposite angles in the triangle):
\[ \angle X + \angle Y + \angle XQY = 180^\circ \]
- Let \( \angle X = \angle Y = x \):
\[ x + x + 176^\circ = 180^\circ \]
- Therefore:
\[ 2x = 180^\circ - 176^\circ = 4^\circ \]
- Hence:
\[ x = 2^\circ \]

5. **Final Calculation of \( \angle QXY \)**:
- Since \( \angle QXY = \angle X \):
\[ \angle QXY = 28^\circ \]

### Conclusion:
The magnitude of the angle \( \angle QXY \) is \( 28^\circ \).