A `DeltaABC` can be constructed in which `angleB = 60^(@), angleC = 45^(@) and AB + BC + CA = 12 cm`.

Explanation

To construct triangle ABC with the given conditions, follow these steps:

### Step 1: Calculate Angle A
We know that the sum of angles in a triangle is 180 degrees. Therefore, we can find angle A as follows:

\[ \text{Angle A} = 180^\circ - \text{Angle B} - \text{Angle C} \]
\[ \text{Angle A} = 180^\circ - 60^\circ - 45^\circ = 75^\circ \]

### Step 2: Assign Lengths to the Sides
Let the lengths of the sides opposite to angles A, B, and C be denoted as:
- \( a = BC \)
- \( b = CA \)
- \( c = AB \)

According to the problem, we have:
\[ a + b + c = 12 \text{ cm} \]

### Step 3: Use the Angle-Side Ratio
To maintain the angles while ensuring the perimeter is 12 cm, we can assign proportional lengths to the sides based on the angles. We can use the sine rule or simply assign lengths that satisfy the angle ratio.

Let’s assume:
- \( a = k \cdot \sin(75^\circ) \)
- \( b = k \cdot \sin(60^\circ) \)
- \( c = k \cdot \sin(45^\circ) \)

Where \( k \) is a constant that we will determine.

### Step 4: Calculate the Sine Values
Using approximate values for the sine functions:
- \( \sin(75^\circ) \approx 0.9659 \)
- \( \sin(60^\circ) \approx 0.8660 \)
- \( \sin(45^\circ) \approx 0.7071 \)

### Step 5: Set Up the Equation
Now, we can express the perimeter in terms of \( k \):
\[ k \cdot (\sin(75^\circ) + \sin(60^\circ) + \sin(45^\circ)) = 12 \]
\[ k \cdot (0.9659 + 0.8660 + 0.7071) = 12 \]
\[ k \cdot 2.539 = 12 \]
\[ k = \frac{12}{2.539} \approx 4.73 \]

### Step 6: Calculate the Lengths of the Sides
Now we can find the lengths of the sides:
- \( a = 4.73 \cdot 0.9659 \approx 4.57 \text{ cm} \)
- \( b = 4.73 \cdot 0.8660 \approx 4.10 \text{ cm} \)
- \( c = 4.73 \cdot 0.7071 \approx 3.34 \text{ cm} \)

### Step 7: Construct the Triangle
1. Draw a line segment \( BC \) of length \( a \).
2. At point \( B \), construct an angle of \( 60^\circ \) using a protractor.
3. At point \( C \), construct an angle of \( 45^\circ \).
4. Extend the lines until they intersect at point \( A \).
5. Measure and adjust the lengths of sides \( AB \) and \( AC \) to ensure they are approximately \( c \) and \( b \) respectively.

### Step 8: Verify the Construction
Check that the sum of the lengths \( a + b + c = 12 \) cm and that the angles are \( 60^\circ \), \( 45^\circ \), and \( 75^\circ \).