Which of the following statements is false ?

Options

A. A quadrilateral has four sides and four vertices
B. A quadrilateral has four angles
C. A quadrilateral has four diagonals
D. A quadrilateral has two diagonals

Answer

To determine which statement about quadrilaterals is false, let's analyze each statement one by one.

### Step-by-Step Solution:

1. **Understanding Quadrilaterals**:
A quadrilateral is defined as a polygon with four sides. This means that any shape with four edges is classified as a quadrilateral.

**Hint**: Remember that the definition of a quadrilateral is key to understanding the statements.

2. **Analyzing Statement 1**:
The first statement claims that "a quadrilateral has four sides and four vertices."
- A quadrilateral indeed has four sides (e.g., in a square or rectangle).
- It also has four vertices (corners).
- Therefore, this statement is **true**.

**Hint**: Count the sides and vertices of a simple quadrilateral to verify this statement.

3. **Analyzing Statement 2**:
The second statement states that "a quadrilateral has four angles."
- A quadrilateral has four angles, one at each vertex.
- For example, in a rectangle, the angles are 90 degrees each, and there are four of them.
- Thus, this statement is also **true**.

**Hint**: Visualize or draw a quadrilateral to see the angles formed at each vertex.

4. **Analyzing Statement 3**:
The third statement claims that "a quadrilateral has four diagonals."
- To find the number of diagonals in a quadrilateral, we can use the formula:
\[ \text{Number of diagonals} = \frac{n(n-3)}{2} \]
where \( n \) is the number of sides. For a quadrilateral, \( n = 4 \):
\[ \text{Number of diagonals} = \frac{4(4-3)}{2} = \frac{4 \times 1}{2} = 2 \]
- Therefore, a quadrilateral has **two diagonals**, not four. This statement is **false**.

**Hint**: Use the diagonal formula to calculate the number of diagonals in different polygons.

5. **Analyzing Statement 4**:
The fourth statement says "a quadrilateral has two diagonals."
- As calculated above, a quadrilateral indeed has two diagonals.
- Hence, this statement is **true**.

**Hint**: Recall the diagonal definition and how they connect non-adjacent vertices.

### Conclusion:
The false statement among the options provided is: **"A quadrilateral has four diagonals."**