Which of the following statements is not correct ?

Options

A. The sum of any two sides of a triangle is greater than the third side.
B. A triangle can have all its angles acute.
C. A right angles triangle cannot be equilateral.
D. Difference of any two sides of a triangle is greater than the third side.

Answer

To determine which of the statements is not correct, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement:** The sum of any two sides of a triangle is greater than the third side.

**Explanation:** This is a fundamental property of triangles known as the Triangle Inequality Theorem. For any triangle with sides \(a\), \(b\), and \(c\), the following must hold true:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)

**Conclusion:** This statement is correct.

### Step 2: Analyze the second statement
**Statement:** A triangle can have all its angles acute.

**Explanation:** An equilateral triangle, where all sides are equal, has angles of \(60^\circ\). Since \(60^\circ\) is less than \(90^\circ\), all angles in an equilateral triangle are acute.

**Conclusion:** This statement is correct.

### Step 3: Analyze the third statement
**Statement:** A right angle triangle cannot be an equilateral triangle.

**Explanation:** A right angle triangle has one angle equal to \(90^\circ\). In contrast, an equilateral triangle has all angles equal to \(60^\circ\). Therefore, it is impossible for a triangle to be both a right angle triangle and an equilateral triangle at the same time.

**Conclusion:** This statement is correct.

### Step 4: Analyze the fourth statement
**Statement:** The difference of any two sides of a triangle is greater than the third side.

**Explanation:** This statement is incorrect. According to the Triangle Inequality Theorem, the difference of two sides of a triangle must be less than the third side. For example, if we take two sides \(a\) and \(b\) of a triangle, the difference \(a - b\) can be less than or equal to the third side \(c\).

**Example:** If \(a = 5\) and \(b = 4\), then \(a - b = 1\), which is not greater than \(c = 3\).

**Conclusion:** This statement is not correct.

### Final Answer
The statement that is not correct is: **The difference of any two sides of a triangle is greater than the third side.**

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