The degree measure of each of the three angles of a triangle is an integer. Which of the following could NOT be the ratio of their measures ?

Options

A. `2:3: 4`
B. `3:4:5`
C. `5:6:7`
D. `6:7:8`

Answer

To determine which of the given ratios could NOT be the ratio of the measures of the angles in a triangle, we need to remember that the sum of the angles in a triangle is always 180 degrees. We will analyze each ratio step by step.

### Step 1: Understanding the Ratios
We are given several ratios of angles. For each ratio, we will express the angles in terms of a variable \( x \) and then check if the angles are integers.

### Step 2: Analyze Each Ratio

1. **Ratio 2:3:4**
- Angles: \( 2x, 3x, 4x \)
- Sum: \( 2x + 3x + 4x = 9x \)
- Set equal to 180 degrees:
\[ 9x = 180 \implies x = 20 \]
- Angles:
- \( 2x = 40^\circ \)
- \( 3x = 60^\circ \)
- \( 4x = 80^\circ \)
- All angles are integers.

2. **Ratio 3:4:5**
- Angles: \( 3x, 4x, 5x \)
- Sum: \( 3x + 4x + 5x = 12x \)
- Set equal to 180 degrees:
\[ 12x = 180 \implies x = 15 \]
- Angles:
- \( 3x = 45^\circ \)
- \( 4x = 60^\circ \)
- \( 5x = 75^\circ \)
- All angles are integers.

3. **Ratio 5:6:7**
- Angles: \( 5x, 6x, 7x \)
- Sum: \( 5x + 6x + 7x = 18x \)
- Set equal to 180 degrees:
\[ 18x = 180 \implies x = 10 \]
- Angles:
- \( 5x = 50^\circ \)
- \( 6x = 60^\circ \)
- \( 7x = 70^\circ \)
- All angles are integers.

4. **Ratio 6:7:8**
- Angles: \( 6x, 7x, 8x \)
- Sum: \( 6x + 7x + 8x = 21x \)
- Set equal to 180 degrees:
\[ 21x = 180 \implies x = \frac{180}{21} \approx 8.57 \]
- Angles:
- \( 6x \approx 51.43^\circ \)
- \( 7x \approx 59.99^\circ \)
- \( 8x \approx 68.57^\circ \)
- These angles are not integers.

### Conclusion
The ratio **6:7:8** could NOT be the ratio of the measures of the angles in a triangle since it does not yield integer angles.

---