Question

ab and cd are parallel lines.
which translation of the plane can we use to prove angles x and y are congruent, and why?
choose 1 answer:
a a translation along the directed line segment ab maps line cd onto line ab and angle y onto angle x.
b a translation along the directed line segment ac maps line ab onto line cd and angle x onto angle y.
c a translation along the directed line segment cb maps line cd onto line ab and angle y onto angle x.

Explanation:

To determine the correct translation, we analyze each option:

• Option A: Translating along \( \overrightarrow{AB} \) would not map \( \overleftrightarrow{CD} \) to \( \overleftrightarrow{AB} \) appropriately as the direction and segment length don't align for this mapping.
• Option B: Translating along \( \overrightarrow{AC} \) (where \( A \) and \( C \) are corresponding points on the parallel lines) will map \( \overleftrightarrow{AB} \) onto \( \overleftrightarrow{CD} \) (since \( AB \parallel CD \)) and angle \( x \) onto angle \( y \) because translation preserves angle measure and maps the lines and angles consistently.
• Option C: Translating along \( \overrightarrow{CB} \) does not correctly map \( \overleftrightarrow{CD} \) to \( \overleftrightarrow{AB} \) in a way that aligns angle \( y \) to \( x \) as the segment direction and length don't support this.

Answer:

B. A translation along the directed line segment \( \boldsymbol{AC} \) maps line \( \overleftrightarrow{AB} \) onto line \( \overleftrightarrow{CD} \) and angle \( \boldsymbol{x} \) onto angle \( \boldsymbol{y} \).