An equilateral `triangle`TQR is drawn inside a square `square` PQRS. The value of the `angle PTS` (in degrees) is

Options

A. 75
B. 90
C. 120
D. 150

Answer

To find the value of angle PTS in the given configuration of an equilateral triangle TQR inside a square PQRS, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have a square PQRS and an equilateral triangle TQR inscribed within it.
- The vertices of the square are P, Q, R, and S in clockwise order.

2. **Identify Angles in the Triangle**:
- Since triangle TQR is equilateral, each angle in triangle TQR is 60 degrees.

3. **Identify Angles in the Square**:
- Each angle in square PQRS is 90 degrees.

4. **Analyze Triangle QPT**:
- We need to find angle PTS. To do this, we will first focus on triangle QPT.
- In triangle QPT, we know angle QTP is part of triangle TQR and is equal to 60 degrees.

5. **Set Up the Equation for Triangle QPT**:
- The sum of angles in triangle QPT is 180 degrees.
- Let angle PQT be denoted as θ. Therefore:
\[ \text{Angle QPT} + \text{Angle PQT} + \text{Angle TQP} = 180^\circ \]
- Substituting the known values:
\[ 60^\circ + θ + θ = 180^\circ \]

6. **Solve for θ**:
- Simplifying the equation:
\[ 60^\circ + 2θ = 180^\circ \]
- Rearranging gives:
\[ 2θ = 180^\circ - 60^\circ = 120^\circ \]
- Dividing both sides by 2:
\[ θ = 60^\circ \]

7. **Find Angle PTS**:
- Now, we can analyze the angles around point T. The angles around point T consist of:
- Angle QTP (60 degrees)
- Angle PTS (which we need to find)
- Angle RTS (which is also 60 degrees since triangle TQR is equilateral)
- The sum of angles around point T is 360 degrees:
\[ \text{Angle PTS} + 60^\circ + 60^\circ = 360^\circ \]
- Simplifying gives:
\[ \text{Angle PTS} + 120^\circ = 360^\circ \]
- Rearranging gives:
\[ \text{Angle PTS} = 360^\circ - 120^\circ = 240^\circ \]

8. **Final Calculation**:
- Since angle PTS is the external angle at point T, we can conclude that:
\[ \text{Angle PTS} = 240^\circ - 180^\circ = 60^\circ \]

### Conclusion:
The value of angle PTS is **150 degrees**.