Construction of Quadrilateral ABCD:

To construct quadrilateral ABCD with the given measurements, follow the steps below:

1. Draw a line segment AB of length 4 cm.
2. From point B, draw a line segment BC of length 3 cm.
3. At point C, draw a line segment CD of length 4 cm such that it forms an angle with BC.
4. From point D, draw a line segment DA of length 3 cm such that it forms an angle with CD.
5. Join points A and C to form diagonal AC.

Calculating the Measure of Diagonal BD:

To find the measure of diagonal BD, we can use the property of a parallelogram. In a parallelogram, opposite sides are equal in length.

1. In quadrilateral ABCD, AB and CD are opposite sides.
- AB = 4 cm (given)
- CD = 4 cm (given)

2. Since AB = CD, we can conclude that quadrilateral ABCD is a parallelogram.

3. In a parallelogram, diagonals bisect each other. Therefore, diagonal AC bisects diagonal BD.

4. Given that AC = 5 cm, we can determine the measure of half of diagonal BD.

- AC = BD/2
- 5 = BD/2
- BD = 2 * 5
- BD = 10 cm

Thus, the measure of diagonal BD is 10 cm.

Determining the Measure of Angle A:

To find the measure of angle A, we can use the Law of Cosines.

1. In triangle ABC, we have the following measurements:
- AB = 4 cm (given)
- BC = 3 cm (given)
- AC = 5 cm (given)

2. Using the Law of Cosines formula:
- AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(A)
- 5^2 = 4^2 + 3^2 - 2 * 4 * 3 * cos(A)
- 25 = 16 + 9 - 24 * cos(A)
- 25 = 25 - 24 * cos(A)
- 24 * cos(A) = 0
- cos(A) = 0
- A = 90 degrees

Therefore, the measure of angle A in quadrilateral ABCD is 90 degrees.

Summary:

In the constructed quadrilateral ABCD, we have:
- AB = 4 cm
- BC = 3 cm
- CD = 4 cm
- AD = 3 cm
- AC = 5 cm
- BD = 10 cm (measure of diagonal)
- Angle A = 90 degrees

The given measurements and construction steps ensure that all the conditions are satisfied, resulting in a quadrilateral ABCD with the desired properties.