Question

using any tools, determine if it is possible to construct a unique figure for each of the following. if the construction is not possible, explain why. complete parts (a) through (c) below. a. a parallelogram, given two of its adjacent sides b. a rhombus, given its diagonals c. a parallelogram given a side and all the angles a. select the correct answer below a. it is not possible. without the angle between the sides, there are infinitely many parallelograms that could be constructed from the two given sides b. it is possible. construct one side that is parallel to one of the given sides. bisect the angle of the new vertex. extend a line from the edge of the new side that is parallel to the other second given side, then find the intersection of that line to the bisecting line. the intersection point of the parallel line and the bisecting line is the second new vertex. connect the new vertices to complete construction c. it is not possible. without the opposite side also being given, there are infinitely many parallelograms that could be constructed from the two given sides d. it is possible. given two sides, any quadrilateral can be constructed

Explanation:

Step1: Analyze parallelogram construction with two adjacent sides

For a parallelogram, knowing only two adjacent sides is not enough. The shape of the parallelogram depends on the angle between these two sides. Without the angle information, we can rotate one side relative to the other to form infinitely - many parallelograms.

Step2: Analyze rhombus construction with diagonals

The diagonals of a rhombus bisect each other at right - angles. Given the lengths of the diagonals, we can construct the perpendicular bisectors of the diagonals and then connect the endpoints to form a unique rhombus.

Step3: Analyze parallelogram construction with a side and all angles

In a parallelogram, opposite angles are equal and adjacent angles are supplementary. Given one side and all the angles, we can construct the other sides based on the angle - side relationships, resulting in a unique parallelogram.

Answer:

a. A. It is not possible. Without the angle between the sides, there are infinitely many parallelograms that could be constructed from the two given sides
b. It is possible. The diagonals of a rhombus bisect each other at right - angles. Given their lengths, we can construct the rhombus.
c. It is possible. Given a side and all the angles, we can construct the parallelogram uniquely.