To find the length of CD in the given quadrilateral ABCD, we can use the property that opposite angles of a quadrilateral are equal.

Given:
AB = 4 cm

Solution:
Let's label the opposite angles of the quadrilateral as ∠A and ∠C, and the sides opposite to these angles as AD and BC, respectively.

According to the property, ∠A = ∠C.

Now, we can use the property that the sum of angles in a quadrilateral is 360 degrees.

∠A + ∠B + ∠C + ∠D = 360°

Since ∠A = ∠C, we can rewrite the equation as:

∠A + ∠B + ∠A + ∠D = 360°

2∠A + ∠B + ∠D = 360°

But opposite angles are equal, so ∠A = ∠C = ∠B = ∠D

Substituting this into the equation:

2∠A + ∠A + ∠A = 360°

4∠A = 360°

∠A = 360° / 4

∠A = 90°

Now, we can use the property of a right-angled triangle to find the length of CD.

In triangle ABC, we have a right angle at B.

Using the Pythagorean theorem, we can find the length of BC:

BC² = AB² + AC²

BC² = 4² + AC²

BC² = 16 + AC²

Since opposite angles are equal, ∠A = ∠C = 90°.

In triangle ACD, we have a right angle at A.

Using the Pythagorean theorem, we can find the length of AC:

AC² = AD² + CD²

Since ∠A = ∠C = 90°, AD and CD are the legs of a right-angled triangle.

But AD = BC (opposite sides of a quadrilateral are equal), so we can substitute BC for AD in the equation:

AC² = BC² + CD²

AC² = 16 + CD²

Now, we know that AC = 4 cm and AC² = 16 + CD².

Substituting the values in the equation:

4² = 16 + CD²

16 = 16 + CD²

CD² = 16 - 16

CD² = 0

Taking the square root on both sides:

CD = √0

CD = 0

Therefore, the length of CD is 0 cm.

Since none of the given options match the calculated length, it appears that there may be an error in the question or options provided.