The sum of all the angles of a quadrilateral is

Options

A. `180^(@)`
B. `270^(@)`
C. `360^(@)`
D. `400^(@)`

Answer

To find the sum of all the angles of a quadrilateral, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Quadrilateral**: Let's denote the quadrilateral as ABCD.

2. **Draw a Diagonal**: Draw a diagonal line from vertex A to vertex C. This diagonal divides the quadrilateral into two triangles: triangle ABD and triangle BCD.

3. **Apply the Angle Sum Property of Triangles**:
- For triangle ABD, the sum of the angles is:
\[ \text{Angle A} + \text{Angle B} + \text{Angle D} = 180^\circ \]
- For triangle BCD, the sum of the angles is:
\[ \text{Angle B} + \text{Angle C} + \text{Angle D} = 180^\circ \]

4. **Combine the Angles**: Now, we add the two equations from the triangles:
\[ (\text{Angle A} + \text{Angle B} + \text{Angle D}) + (\text{Angle B} + \text{Angle C} + \text{Angle D}) = 180^\circ + 180^\circ \]

5. **Simplify the Equation**:
- Combine like terms:
\[ \text{Angle A} + 2 \cdot \text{Angle B} + \text{Angle C} + 2 \cdot \text{Angle D} = 360^\circ \]

6. **Recognize the Angles**: Notice that:
- Angle A, Angle B, Angle C, and Angle D are the angles of the quadrilateral ABCD.
- Thus, we can rewrite the equation as:
\[ \text{Angle A} + \text{Angle B} + \text{Angle C} + \text{Angle D} = 360^\circ \]

7. **Conclusion**: Therefore, the sum of all the angles in a quadrilateral is:
\[ \text{Sum of angles} = 360^\circ \]