Given the following data which, can form two triangles?
I. `angleC=30^(@),c=8,b=12`
II. `angleB=45^(@),a=12sqrt(2),b=12sqrt(2)`
III. `angleC=60^(@),b=12,c=5sqrt(3)`

Options

A. only I
B. only II
C. only III
D. only I and II

Explanation

To determine which sets of data can form two triangles, we will follow the conditions for triangle formation based on the given angle and sides. Let's analyze each case step by step.

### Given Data:
1. **I.** \( \angle C = 30^\circ, c = 8, b = 12 \)
2. **II.** \( \angle B = 45^\circ, a = 12\sqrt{2}, b = 12\sqrt{2} \)
3. **III.** \( \angle C = 60^\circ, b = 12, c = 5\sqrt{3} \)

### Step 1: Analyze the first set of data (I)
- Given: \( \angle C = 30^\circ, c = 8, b = 12 \)
- Identify the largest side: \( b = 12 \)
- Calculate the altitude from angle C:
\[ \text{Altitude} = b \cdot \sin(\angle C) = 12 \cdot \sin(30^\circ) = 12 \cdot \frac{1}{2} = 6 \]
- Compare the smallest side \( c = 8 \) with the altitude \( 6 \) and the largest side \( 12 \):
- Since \( 6 < 8 < 12 \), two triangles can be formed.

### Step 2: Analyze the second set of data (II)
- Given: \( \angle B = 45^\circ, a = 12\sqrt{2}, b = 12\sqrt{2} \)
- Identify the largest side: \( a = b = 12\sqrt{2} \)
- Calculate the altitude from angle B:
\[ \text{Altitude} = a \cdot \sin(\angle B) = 12\sqrt{2} \cdot \sin(45^\circ) = 12\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 12 \]
- Compare the smallest side (which is equal to the other side) with the altitude:
- Since \( 12 = 12 \), only one triangle can be formed.

### Step 3: Analyze the third set of data (III)
- Given: \( \angle C = 60^\circ, b = 12, c = 5\sqrt{3} \)
- Identify the largest side: \( b = 12 \)
- Calculate the altitude from angle C:
\[ \text{Altitude} = b \cdot \sin(\angle C) = 12 \cdot \sin(60^\circ) = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \]
- Compare the smallest side \( c = 5\sqrt{3} \) with the altitude \( 6\sqrt{3} \) and the largest side \( 12 \):
- Since \( 5\sqrt{3} < 6\sqrt{3} < 12 \), no triangle can be formed.

### Conclusion:
- Only the first set of data can form two triangles.
- The second set can form one triangle.
- The third set cannot form any triangles.

### Final Answer:
Only the data from **I** can form two triangles.