PQRS is a cyclic quadrilateral and PQ is the diameter of circle . If `angleRPQ = 38^(@)` , what is the value (in degrees) of `anglePSR` ?

Options

A. 52
B. 77
C. 128
D. 142

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of cyclic quadrilaterals
A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. One important property of cyclic quadrilaterals is that the sum of the opposite angles is equal to 180 degrees.

### Step 2: Identify the given information
We are given that PQ is the diameter of the circle and that angle RPQ = 38 degrees. We need to find the value of angle PSR.

### Step 3: Apply the inscribed angle theorem
Since PQ is the diameter of the circle, angle RPQ is an inscribed angle that subtends the arc PS. According to the inscribed angle theorem, any angle inscribed in a semicircle is a right angle. Therefore, angle RQP is 90 degrees.

### Step 4: Find angle QRS
In triangle PQR, we know:
- Angle RPQ = 38 degrees
- Angle RQP = 90 degrees

Using the triangle sum property (the sum of angles in a triangle is 180 degrees), we can find angle QRS:
\[ \text{Angle QRS} = 180 - \text{Angle RPQ} - \text{Angle RQP} \]
\[ \text{Angle QRS} = 180 - 38 - 90 = 52 \text{ degrees} \]

### Step 5: Use the property of cyclic quadrilaterals
Now, since PQRS is a cyclic quadrilateral, we can use the property that the sum of opposite angles is 180 degrees:
\[ \text{Angle PSR} + \text{Angle QRS} = 180 \]
Substituting the known value of angle QRS:
\[ \text{Angle PSR} + 52 = 180 \]

### Step 6: Solve for angle PSR
Now we can solve for angle PSR:
\[ \text{Angle PSR} = 180 - 52 = 128 \text{ degrees} \]

### Final Answer
Thus, the value of angle PSR is **128 degrees**.

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