What is the maximum number of obtuse angles that a quadrilateral can have ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the maximum number of obtuse angles that a quadrilateral can have, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Angles in a Quadrilateral**:
A quadrilateral has four interior angles. The sum of these angles is always 360 degrees.

2. **Definition of Obtuse Angles**:
An obtuse angle is defined as an angle that is greater than 90 degrees but less than 180 degrees.

3. **Maximum Number of Angles**:
Since there are only four angles in a quadrilateral, the maximum number of obtuse angles cannot exceed four.

4. **Testing the Case of Four Obtuse Angles**:
- If we assume all four angles are obtuse, then each angle must be greater than 90 degrees.
- Therefore, the minimum sum of four obtuse angles would be \(90 + 90 + 90 + 90 = 360\) degrees.
- However, since each angle is greater than 90 degrees, the total would exceed 360 degrees, which is not possible.

5. **Testing the Case of Three Obtuse Angles**:
- Now, let’s consider the case where three angles are obtuse.
- Assume each of the three obtuse angles is 100 degrees.
- The sum of these three angles would be \(100 + 100 + 100 = 300\) degrees.
- The fourth angle can then be calculated as \(360 - 300 = 60\) degrees, which is acute.
- This configuration is valid since we have three obtuse angles and one acute angle.

6. **Conclusion**:
Since having four obtuse angles is not possible, and we can have three obtuse angles with one acute angle, we conclude that the maximum number of obtuse angles in a quadrilateral is **3**.

### Final Answer:
The maximum number of obtuse angles that a quadrilateral can have is **3**.

---