PQRS is a parallelogram in which
`angle P=70^@, angle Q=90^@` and `angle R=100^@`. How many points in the plane of the quadrilateral are there such that a point is equidistant from its vertices ?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to analyze the properties of the given quadrilateral PQRS and determine if it can be classified as a parallelogram based on the angles provided.

### Step-by-Step Solution:

1. **Identify the Properties of a Parallelogram**:
- In a parallelogram, opposite angles are equal.
- The sum of the angles in any quadrilateral is 360 degrees.

2. **Given Angles**:
- Angle P = 70 degrees
- Angle Q = 90 degrees
- Angle R = 100 degrees
- We need to find Angle S.

3. **Calculate Angle S**:
- Using the property that the sum of angles in a quadrilateral is 360 degrees:
\[ \text{Angle S} = 360^\circ - (\text{Angle P} + \text{Angle Q} + \text{Angle R}) \]
\[ \text{Angle S} = 360^\circ - (70^\circ + 90^\circ + 100^\circ) = 360^\circ - 260^\circ = 100^\circ \]

4. **Check Opposite Angles**:
- Now we have:
- Angle P = 70 degrees
- Angle Q = 90 degrees
- Angle R = 100 degrees
- Angle S = 100 degrees
- We see that Angle R and Angle S are equal (100 degrees), but Angle P (70 degrees) is not equal to Angle Q (90 degrees).

5. **Conclusion**:
- Since the angles do not satisfy the properties of a parallelogram (specifically, opposite angles must be equal), PQRS cannot be classified as a parallelogram.
- Therefore, there are no points in the plane that are equidistant from the vertices of PQRS.

6. **Final Answer**:
- The number of points in the plane that are equidistant from the vertices of quadrilateral PQRS is **0**.