To determine which of the given equations is a line of reflection that would map the quadrilateral ABCD onto itself, let's examine each equation step-by-step:
1. Equation: [tex]\( y = 1 \)[/tex]
- This is a horizontal line. When reflecting over this line, each point [tex]\((x, y)\)[/tex] would map to [tex]\((x, 2 - y)\)[/tex]. This will not necessarily map ABCD onto itself as all y-coordinates would be altered.
2. Equation: [tex]\( 2x + y = 2 \)[/tex]
- This line is neither vertical nor horizontal. To check if it maps the quadrilateral onto itself, rearrange terms to better understand the reflection behavior. However, in general, this leads to transformations that do not consistently map each vertex of ABCD back onto another vertex of ABCD.
3. Equation: [tex]\( x + y = 4 \)[/tex]
- Similar to the previous one, rewriting as [tex]\(y = 4 - x\)[/tex], the reflection would again involve complex transformations altering both x and y coordinates. This too is unlikely to map all vertices onto themselves consistently.
4. Equation: [tex]\( x + y = 1 \)[/tex]
- This equation suggests a line inclined at 45 degrees. Reflection across this line means that for any point [tex]\((x, y)\)[/tex], transforming it into its image would imply swapping and adjusting coordinates such that the transformed points fit back into the original structure of ABCD. This behavior consistently maintains the structure of ABCD.
Upon analyzing these equations, we conclude that only [tex]\( x + y = 1 \)[/tex] ensures that each vertex of quadrilateral ABCD maps back onto another vertex of ABCD or onto itself consistently.
Thus, the correct answer is:
[tex]\[ x + y = 1 \][/tex]
Hence, the answer to the given multiple-choice question is:
[tex]\[ 4 \][/tex]
