If each angle of a triangle is less than the sum of the other two, then the triangle is

Options

A. obtuse angled
B. right angled
C. acute angled
D. equilateral

Answer

To solve the problem, we need to analyze the condition given for the angles of a triangle. We will denote the angles of the triangle as \( A \), \( B \), and \( C \).

### Step-by-step Solution:

1. **Understanding the Condition**:
The problem states that each angle of the triangle is less than the sum of the other two angles. This can be expressed mathematically as:
- \( A < B + C \)
- \( B < A + C \)
- \( C < A + B \)

2. **Using the Triangle Angle Sum Property**:
We know that the sum of the angles in a triangle is always \( 180^\circ \):
\[ A + B + C = 180^\circ \]

3. **Analyzing Each Angle**:
Let's take the first inequality \( A < B + C \). We can substitute \( B + C \) with \( 180^\circ - A \) (since \( B + C = 180^\circ - A \)):
\[ A < 180^\circ - A \]
Adding \( A \) to both sides gives:
\[ 2A < 180^\circ \]
Dividing both sides by 2 results in:
\[ A < 90^\circ \]

4. **Repeating for Other Angles**:
We can apply the same reasoning to angles \( B \) and \( C \):
- For \( B < A + C \):
\[ B < 180^\circ - B \implies 2B < 180^\circ \implies B < 90^\circ \]
- For \( C < A + B \):
\[ C < 180^\circ - C \implies 2C < 180^\circ \implies C < 90^\circ \]

5. **Conclusion**:
Since all three angles \( A \), \( B \), and \( C \) are less than \( 90^\circ \), we conclude that the triangle is an **acute-angled triangle**.

### Final Answer:
The triangle is an **acute-angled triangle**.

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