Given:
- The triangular sign board is isosceles.
- The unequal side is 7 cm.
- One of the equal sides is 6 cm.

To Find:
The measure of the third side.

Solution:

Since the triangular sign board is isosceles, it means that two sides of the triangle are equal in length. Let's denote the length of the unequal side as 'a' and the length of the equal side as 'b'.

Given that the unequal side is 7 cm and one of the equal sides is 6 cm, we have:
- a = 7 cm
- b = 6 cm

Now, let's determine the measure of the third side using the properties of an isosceles triangle.

Identify:
- The length of the unequal side: a = 7 cm
- The length of one of the equal sides: b = 6 cm

Properties of an Isosceles Triangle:
- An isosceles triangle has two equal sides.
- The angles opposite the equal sides are also equal.

Calculation:

Since the triangle is isosceles, the two equal sides are of length 6 cm each. Let's denote the length of the third side as 'c'.

Since the triangle is isosceles, the two equal angles opposite the equal sides are also equal. Let's denote the measure of each of these angles as 'x'.

Using the properties of an isosceles triangle, we can write:
- a = b = 6 cm
- Angle A = Angle B = x (opposite to the equal sides)
- Angle C (opposite to the unequal side)

Since the sum of angles in a triangle is 180°, we have:
x + x + C = 180°
2x + C = 180°

Since the triangle is isosceles, the sum of the angles opposite the equal sides is equal to 180°. Therefore, we have:
2x + C = 180°

In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, we can also write:
x + x + C = 180°
2x + C = 180°

Since the triangle is isosceles, the sum of the angles opposite the equal sides is equal to 180°. Therefore, we have:
2x + C = 180°

Now, let's substitute the known values into the equation:
2x + 7 = 180°

Solving this equation gives us:
2x = 180° - 7
2x = 173°
x = 173°/2
x = 86.5°

Therefore, the measure of the third angle, C, is:
C = 180° - 2x
C = 180° - 2(86.5°)
C = 180° - 173°
C = 7°

Conclusion:
The measure of the third side, C, is 7°. Therefore, the correct answer is option 'b' - 6 cm.