Let's analyze the given statements regarding the construction of a rhombus and a parallelogram based on specific measurements.

Analyzing Statement I: Rhombus Construction Uniqueness

Statement I says that only one Rhombus ABCD can be drawn with AB = 4 cm and diagonal BD = 5 cm.

• A rhombus is a quadrilateral where all four sides are equal in length.
• Given AB = 4 cm, this means all sides of the rhombus are 4 cm: AB = BC = CD = DA = 4 cm.
• We are also given the length of one diagonal, BD = 5 cm.

Consider the triangle ABD within the rhombus. Its sides are:

• AB = 4 cm (a side of the rhombus)
• AD = 4 cm (a side of the rhombus, adjacent to AB)
• BD = 5 cm (the given diagonal)

We have the lengths of all three sides of triangle ABD: 4 cm, 4 cm, and 5 cm.

According to the SSS (Side-Side-Side) criterion for triangle congruence and construction, if you know the lengths of all three sides of a triangle, you can draw only one unique triangle (up to congruence).

Since triangle ABD is uniquely determined by its side lengths (4 cm, 4 cm, 5 cm), the positions of vertices A, B, and D relative to each other are fixed. Once triangle ABD is fixed, the location of point C in the rhombus is also uniquely determined because ABCD must be a parallelogram (every rhombus is a parallelogram) and all sides must be 4 cm. Point C must be 4 cm away from B and 4 cm away from D. The intersection of arcs with radius 4 cm centered at B and D (on the opposite side of BD from A) will give the unique position for C.

Thus, only one such rhombus can be constructed with the given side and diagonal lengths.

Therefore, Statement I is true.

Analyzing Statement II: Parallelogram Construction Uniqueness

Statement II says that only one parallelogram ABCD can be drawn with AB = 6 cm and diagonal BD = 8 cm.

• A parallelogram is a quadrilateral where opposite sides are equal in length.
• Given AB = 6 cm, this means CD = 6 cm.
• We are also given the length of one diagonal, BD = 8 cm.

Consider the triangle ABD within the parallelogram. Its sides are:

• AB = 6 cm (a side of the parallelogram)
• BD = 8 cm (the given diagonal)
• AD = ? (the adjacent side of the parallelogram, opposite to BC)

To uniquely determine a triangle using side lengths, you need either all three side lengths (SSS) or two sides and the included angle (SAS). In triangle ABD, we only know two sides (AB = 6 cm and BD = 8 cm). The length of the third side, AD, which is the other side of the parallelogram, is not given. The angle between AB and BD (∠ABD) or the angle between AD and BD (∠ADB) or the angle at A (∠BAD) is also not given.

Since the length of the adjacent side AD can be any length (as long as it satisfies the triangle inequality for triangle ABD, i.e., $|\text{AB} - \text{BD}| < \text{AD} < \text{AB} + \text{BD}$, which means $|6-8| < \text{AD} < 6+8$, or $2 < \text{AD} < 14$), different lengths for AD will result in different possible triangles ABD. Each different triangle ABD will form a different parallelogram.

For example, if AD = 5 cm, triangle ABD has sides 6, 8, 5. This forms a parallelogram with sides 6 and 5 and diagonal 8. If AD = 7 cm, triangle ABD has sides 6, 8, 7. This forms a different parallelogram with sides 6 and 7 and diagonal 8.

Since the length of the adjacent side is not fixed, more than one parallelogram can be drawn with the given information (one side and one diagonal).

Therefore, Statement II is false.

Conclusion

Based on the analysis:

• Statement I is true (Only one rhombus can be drawn).
• Statement II is false (More than one parallelogram can be drawn).

This matches the option which states that Statement I is true but Statement II is false.

Revision Table: Geometry Figure Construction

Understanding what information uniquely determines a geometric figure is crucial.

• Triangle: Uniquely determined by SSS, SAS, ASA, AAS, or RHS. Not by SSA.
• Square: Uniquely determined by side length or diagonal length.
• Rhombus: Uniquely determined by side length and one angle, or side length and one diagonal, or two diagonals.
• Rectangle: Uniquely determined by side lengths (length and width) or one side and a diagonal.
• Parallelogram: Uniquely determined by two adjacent side lengths and the included angle, or two adjacent side lengths and a diagonal, or one side length and two diagonals. Not uniquely determined by just one side and one diagonal (as shown in Statement II).

Additional Information: Properties of Rhombus and Parallelogram

Let's review some key properties that are relevant to these constructions.

• Rhombus Properties:
  • All four sides are equal.
  • Opposite angles are equal.
  • Diagonals bisect each other at right angles.
  • Diagonals bisect the angles at the vertices.
  • A rhombus is a type of parallelogram.
• Parallelogram Properties:
  • Opposite sides are equal and parallel.
  • Opposite angles are equal.
  • Consecutive angles are supplementary (add up to 180 degrees).
  • Diagonals bisect each other.

In Statement I, knowing the side length and diagonal BD allows us to form a triangle ABD with sides 4, 4, and 5. The rigidity of this triangle structure (SSS) is key to the uniqueness of the rhombus.

In Statement II, knowing side AB and diagonal BD only provides two sides of the triangle ABD. The length of the adjacent side AD is not fixed, allowing the parallelogram to be "squashed" or "stretched" along the diagonal BD while keeping AB and BD constant, thus leading to multiple possible shapes.