Question

Which set of angle measures can be used to construct an acute scalene triangle?
36°, 57°, 87°
46°, 57°, 67°
46°, 67°, 67°
36°, 47°, 97°

Answer 1

To determine which set of angle measures can be used to construct an acute scalene triangle, we need to ensure that:

1. The measures add up to 180°.
2. All angle measures are less than 90° (to ensure the triangle is acute).
3. All angle measures are different (to ensure the triangle is scalene).

Let's analyze each set of angle measures:

1. **36°, 57°, 87°**
- Sum: \(36 + 57 + 87 = 180°\)
- All angles are less than 90° (acute).
- Angles are different (scalene).
- **This set can construct an acute scalene triangle.**

2. **46°, 57°, 67°**
- Sum: \(46 + 57 + 67 = 170°\) (not equal to 180°).
- This fails criterion 1; thus, it cannot form a triangle.

3. **46°, 67°, 67°**
- Sum: \(46 + 67 + 67 = 180°\)
- Not all angles are different (two angles are 67°), so it cannot be scalene. This fails criterion 3.

4. **36°, 47°, 97°**
- Sum: \(36 + 47 + 97 = 180°\)
- One angle (97°) is greater than 90°, so it is not acute. This fails criterion 2.

The only set that satisfies all conditions for constructing an acute scalene triangle is:

**36°, 57°, 87°**.