The side QR of an equilateral triangle PQR is produced to the point S in such a way that QR = RS and P is joined to S. Then the measure of `/_PSR` is

Options

A. `30^@`
B. `15^@`
C. `60^@`
D. `45^@`

Answer

To find the measure of angle \( \angle PSR \) in the given problem, we will follow these steps:

### Step 1: Draw the Equilateral Triangle
First, we need to draw an equilateral triangle \( PQR \). In an equilateral triangle, all sides are equal and all angles are \( 60^\circ \).

### Step 2: Extend Side QR to Point S
Next, extend the side \( QR \) to a point \( S \) such that \( QR = RS \). This means that the length of \( RS \) is equal to the length of \( QR \).

### Step 3: Identify the Angles
Since \( PQR \) is an equilateral triangle, we know:
- \( \angle PQR = 60^\circ \)
- \( \angle QRP = 60^\circ \)
- \( \angle RPQ = 60^\circ \)

### Step 4: Use the External Angle Theorem
According to the external angle theorem, the measure of an external angle (in this case, \( \angle PSR \)) is equal to the sum of the measures of the two opposite internal angles. Here, the internal angles are \( \angle PQR \) and \( \angle QRP \).

### Step 5: Calculate \( \angle PSR \)
Since \( \angle PSR \) is the external angle at point \( S \):
\[ \angle PSR = \angle PQR + \angle QRP \]
Substituting the values:
\[ \angle PSR = 60^\circ + 60^\circ = 120^\circ \]

### Step 6: Conclusion
Thus, the measure of \( \angle PSR \) is \( 120^\circ \).

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