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-angled triangle cannot be equilateral
D. Difference of any two sides of a triangle is greater than the third side

Answer

To determine which statement is not correct, let's analyze each option step by step.

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

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

Since this is a fundamental property of triangles, this statement is **true**.

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

**Explanation:** An acute triangle is defined as a triangle where all three angles are less than 90 degrees. An equilateral triangle, for example, has three angles of 60 degrees, which are all acute. Therefore, this statement is also **true**.

### Step 3: Analyze Option 3
**Statement:** A right angle triangle cannot be equilateral.

**Explanation:** A right angle triangle has one angle that is exactly 90 degrees. An equilateral triangle has all three angles equal, which means each angle would be 60 degrees. Since it is impossible for a triangle to have both a right angle (90 degrees) and all angles equal (60 degrees), this statement is **true**.

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

**Explanation:** This statement is incorrect. The correct relationship is that the difference of any two sides of a triangle must be **less than** the third side. For example, if we have a triangle with sides \(a\), \(b\), and \(c\), the following must hold true:
- \(|a - b| < c\)
- \(|a - c| < b\)
- \(|b - c| < a\)

Thus, this statement is **false**.

### Conclusion
The statement that is **not correct** is **Option 4**: "The difference of any two sides of a triangle is greater than the third side."

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