Can all the four angles of a quadrilateral be obtuse angles ? Give reason for your answer.

Options

A. yes
B. no
C. can not determine
D. none of these

Explanation

To determine whether all four angles of a quadrilateral can be obtuse angles, we can analyze the properties of angles in a quadrilateral.

### Step-by-Step Solution:

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

2. **Sum of Angles in a Quadrilateral**: The sum of all interior angles in a quadrilateral is always 360 degrees.

3. **Assuming All Angles are Obtuse**: Let's assume that all four angles (A, B, C, and D) of the quadrilateral are obtuse. Therefore, we can express this as:
- Angle A > 90 degrees
- Angle B > 90 degrees
- Angle C > 90 degrees
- Angle D > 90 degrees

4. **Adding the Angles**: If we add these inequalities together, we get:
\[ A + B + C + D > 90 + 90 + 90 + 90 = 360 \text{ degrees} \]
This means:
\[ A + B + C + D > 360 \text{ degrees} \]

5. **Contradiction**: However, we know that the sum of the angles in any quadrilateral must equal exactly 360 degrees. Therefore, having all four angles greater than 90 degrees leads to a contradiction because it implies that the sum of the angles exceeds 360 degrees.

6. **Conclusion**: Since it is not possible for the sum of the angles to exceed 360 degrees, we conclude that it is impossible for all four angles of a quadrilateral to be obtuse angles.

### Final Answer:
No, all four angles of a quadrilateral cannot be obtuse angles because if they were, their sum would exceed 360 degrees, which contradicts the property that the sum of angles in a quadrilateral is always 360 degrees.