Question

determine whether each claim about the properties of abcd and abcd is true or false.
\\( \overline { a d } \\) and \\( \overline { a ^ { \prime } d ^ { \prime } } \\) are on distinct parallel lines.
\\( \overline { c d } \\) and \\( \overline { c ^ { \prime } d ^ { \prime } } \\) are both perpendicular to the x-axis.

Explanation:

Step1: Analyze the first claim

When a figure is transformed (assuming a translation, rotation, reflection etc. which is common in such problems), if two line - segments are parallel in the original and transformed figure, we check their slopes.
Let's assume \(A(-1, - 1)\), \(D(2,-4)\). The slope of \(\overline{AD}\) is \(m_{AD}=\frac{-4 + 1}{2+1}=\frac{-3}{3}=-1\). If \(A'\) and \(D'\) are the transformed points (assuming a rigid transformation), the slope of \(\overline{A'D'}\) will be the same as the slope of \(\overline{AD}\) (since rigid transformations preserve slope). Parallel lines have the same slope. If they are on distinct lines (not coincident), then \(\overline{AD}\) and \(\overline{A'D'}\) are on distinct parallel lines.

Step2: Analyze the second claim

For a line - segment to be perpendicular to the \(x\) - axis, it must be a vertical line. A vertical line has an undefined slope (or in terms of coordinates, if two points \((x_1,y_1)\) and \((x_2,y_2)\) form a line - segment, then \(x_1=x_2\)).
For \(\overline{CD}\), if \(C(2,1)\) and \(D(2,-4)\), \(x\) - coordinate of \(C\) and \(D\) is \(x = 2\). For \(\overline{C'D'}\) (after a rigid transformation which preserves the \(x\) - coordinate relationship), if \(C'\) and \(D'\) are the images of \(C\) and \(D\) under a rigid transformation (translation, rotation, reflection), the \(x\) - coordinate of \(C'\) and \(D'\) will be the same. A line - segment with the same \(x\) - coordinate for its endpoints is perpendicular to the \(x\) - axis.

Answer:

1. True
2. True