Which of the following group of angles form an acute-angled triangle ?

Options

A. `30^(@), 90^(@), 60^(@)`
B. `120^(@), 30^(@), 30^(@)`
C. `40^(@), 70^(@), 70^(@)`
D. `20^(@), 60^(@), 100^(@)`

Answer

To determine which group of angles forms an acute-angled triangle, we need to follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Acute-Angled Triangle**:
An acute-angled triangle is a triangle where all three angles are less than 90 degrees.

2. **Check the Sum of Angles**:
The sum of the angles in any triangle must equal 180 degrees. Therefore, for a set of angles to form a triangle, they must add up to 180 degrees.

3. **Evaluate Each Option**:
We will analyze each option provided to see if they meet the criteria for an acute-angled triangle.

- **Option 1: 30°, 90°, 60°**
- Check if all angles are less than 90°: 30° is less than 90°, 90° is equal to 90°, and 60° is less than 90°.
- Since one angle is 90°, this is a right-angled triangle, **not acute**.

- **Option 2: 120°, 30°, 30°**
- Check if all angles are less than 90°: 120° is greater than 90°, and both 30° angles are less than 90°.
- Since one angle is greater than 90°, this is an obtuse-angled triangle, **not acute**.

- **Option 3: 40°, 70°, 70°**
- Check if all angles are less than 90°: 40° is less than 90°, 70° is less than 90°, and the other 70° is also less than 90°.
- All angles are less than 90°, and they add up to 180° (40° + 70° + 70° = 180°). This forms an acute-angled triangle.

- **Option 4: 20°, 60°, 100°**
- Check if all angles are less than 90°: 20° is less than 90°, 60° is less than 90°, but 100° is greater than 90°.
- Since one angle is greater than 90°, this is an obtuse-angled triangle, **not acute**.

4. **Conclusion**:
The only option that forms an acute-angled triangle is **Option 3: 40°, 70°, 70°**.