PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If `angleQPR=67^(@)` and `angleSPR=72^(@)`, then `angleQRS=` ___________

Options

A. `41^(@)`
B. `23^(@)`
C. `67^(@)`
D. `18^(@)`

Answer

To solve for the angle \( \angle QRS \) in the cyclic quadrilateral \( PQRS \) where \( PR \) is the diameter of the circle, we can follow these steps:

### Step 1: Understand the properties of cyclic quadrilaterals
In a cyclic quadrilateral, the sum of the opposite angles is \( 180^\circ \). Therefore, we have:
\[ \angle QPS + \angle QRS = 180^\circ \]

### Step 2: Identify the angles given in the question
We are given:
- \( \angle QPR = 67^\circ \)
- \( \angle SPR = 72^\circ \)

### Step 3: Find \( \angle QPS \)
Since \( PR \) is the diameter, we can use the property of angles subtended by a diameter. The angle \( \angle QPS \) can be found using:
\[ \angle QPS = \angle QPR + \angle SPR \]
Substituting the values:
\[ \angle QPS = 67^\circ + 72^\circ = 139^\circ \]

### Step 4: Use the property of cyclic quadrilaterals to find \( \angle QRS \)
Now, we can substitute \( \angle QPS \) into the equation for opposite angles:
\[ \angle QPS + \angle QRS = 180^\circ \]
Substituting \( \angle QPS = 139^\circ \):
\[ 139^\circ + \angle QRS = 180^\circ \]

### Step 5: Solve for \( \angle QRS \)
To find \( \angle QRS \), we rearrange the equation:
\[ \angle QRS = 180^\circ - 139^\circ = 41^\circ \]

### Final Answer
Thus, the value of \( \angle QRS \) is:
\[ \angle QRS = 41^\circ \]

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