Question

quadrilateral abcd is a rhombus. determine which of the statements is a valid explanation or is not a valid explanation for how you know angle b maps onto angle d. translating a copy of the figure down 3 units. select one translating a copy of the figure 5 units to the right. select one rotating a copy of the figure 90 degrees clockwise around the intersection of the two diagonals. select one rotating a copy of the figure 180 degrees about the midpoint of the diagonals. select one reflecting a copy of the figure over bd select one reflecting a copy of the figur

Explanation:

Step1: Recall properties of a rhombus

A rhombus has opposite - angles equal. To map angle B onto angle D, we need a transformation that preserves the shape and orientation in a way that aligns these opposite angles.

Step2: Analyze translations

Translating a copy of the figure down 3 units or 5 units to the right (first two options) only moves the figure without changing its orientation relative to itself. These translations do not map angle B onto angle D as they do not account for the symmetry about the center or axes of the rhombus.

Step3: Analyze 90 - degree rotation

Rotating a copy of the figure 90 degrees clockwise around the intersection of the two diagonals (third option) will not map angle B onto angle D because a 90 - degree rotation in a rhombus does not align opposite angles in this case.

Step4: Analyze 180 - degree rotation

Rotating a copy of the figure 180 degrees about the mid - point of the diagonals (fourth option) will map angle B onto angle D. This is because a 180 - degree rotation about the center of a rhombus (mid - point of diagonals) maps each vertex to its opposite vertex and thus maps one opposite angle onto the other.

Step5: Analyze reflection over BD

Reflecting a copy of the figure over $\overline{BD}$ (fifth option) will map angle B onto itself, not onto angle D.

Answer:

Translating a copy of the figure down 3 units: Not a valid explanation
Translating a copy of the figure 5 units to the right: Not a valid explanation
Rotating a copy of the figure 90 degrees clockwise around the intersection of the two diagonals: Not a valid explanation
Rotating a copy of the figure 180 degrees about the mid - point of the diagonals: Valid explanation
Reflecting a copy of the figure over $\overline{BD}$: Not a valid explanation