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 c. select the correct answer below a. it is possible. given the original side and an angle, construct an adjacent side. bisect the new vertex, then construct a line segment from the first free - edge that intersects the bisecting line. construct the final side, which is parallel to the first constructed edge b. it is possible. construct parallel lines that satisfy the angles for the original side. connect the midpoint of the original side and the intersection of two lines may bisect the new vertices. connect the remaining edges c. it is not possible. there is no unique parallelogram because the other pair of congruent sides can be of any length d. 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

Explanation:

Step1: Analyze part a

A parallelogram given two of its adjacent sides is not sufficient to construct a unique parallelogram. Without an included - angle, we can vary the angle between the two given sides, resulting in infinitely many parallelograms.

Step2: Analyze part b

A rhombus given its diagonals is possible to construct uniquely. The diagonals of a rhombus are perpendicular bisectors of each other. We can first construct two perpendicular segments bisecting each other with lengths equal to the given diagonals, and then connect the endpoints to form the rhombus.

Step3: Analyze part c

A parallelogram given a side and all the angles is not possible to construct a unique parallelogram. Since the opposite sides of a parallelogram are equal, without knowing the length of the other non - given side, we can have infinitely many parallelograms with the given side and angles.

Answer:

a. It is not possible. Without an included angle between the two given adjacent sides, there are infinitely many parallelograms.
b. It is possible. Construct two perpendicular segments bisecting each other and congruent to the given diagonals, then connect the endpoints.
c. It is not possible. There is no unique parallelogram because the other non - given side can be of any length while maintaining the given side and angles.