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.
b. select the correct answer below.
a. 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
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 at least one side given, there are infinitely many rhombuses that could be constructed from the given diagonals
d. it is possible. construct two perpendicular segments bisecting each other and congruent to the given diagonals
e. it is not possible. without at least two adjacent sides and one angle given, there are infinitely many rhombuses that could be constructed from the given diagonals
f. it is possible. connect the edge of each line - segment to the nearest edge, going counter - clockwise. these four line segments complete the construction

Explanation:

Step1: Analyze part (a)

Given two adjacent sides of a parallelogram, without information about angles or the opposite - sides, there are infinitely many parallelograms that can be formed. So, it is not possible to construct a unique parallelogram.

Step2: Analyze part (b)

The diagonals of a rhombus bisect each other at right - angles. Given the lengths of the diagonals of a rhombus, we can construct two perpendicular line segments that bisect each other and are congruent to the given diagonals. Then, by connecting the endpoints of these segments, we can construct a unique rhombus.

Step3: Analyze part (c)

Given a side and all the angles of a parallelogram, since opposite angles of a parallelogram are equal and adjacent angles are supplementary, but without the length of the other side, there are infinitely many parallelograms that can be constructed. So, it is not possible to construct a unique parallelogram.

Answer:

a. 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
b. B. It is possible. Construct two perpendicular segments bisecting each other and congruent to the given diagonals
c. It is not possible. Without the length of the other non - parallel side, there are infinitely many parallelograms that could be constructed from the given side and angles