Question

problem 7: (first taught in lesson 24)
from this given statement, select the definition, property, postulate, or theorem that justifies the prove statement.
given: $overline{uv} \parallel \overline{xy}$
prove: $\angle 1$ and $\angle 2$ are supplementary.
diagram: quadrilateral uvyx with vertices u, v, y, x; $\angle 1$ at u, $\angle 2$ at x
options (a): a if two parallel lines are cut by a transversal, then their corresponding angles are congruent.

Explanation:

1. Identify the lines and transversal: \(UV\parallel XY\) and transversal \(UX\) creates \(\angle1\) and \(\angle2\).
2. Recall angle relationships: For parallel lines cut by a transversal, consecutive interior angles (like \(\angle1\) and \(\angle2\) here) are supplementary (sum to \(180^\circ\)).
3. Relate to options: The given option A is about corresponding angles (congruent, not supplementary), so the correct principle (not fully shown in the provided options) is the "consecutive interior angles supplementary" theorem for parallel lines.

Answer:

(Note: Since the full options B - E are not provided, but based on the context, the correct concept should be "If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary" (assuming the lines \(UV\parallel XY\) and transversal \(UX\), making \(\angle1\) and \(\angle2\) consecutive interior angles). If we consider the given option A is about corresponding angles (which is not for supplementary consecutive angles), but if we assume the intended correct principle is the consecutive interior angles theorem: when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. So the justification is that theorem. However, since only option A is partially shown, but the key idea is the consecutive interior angles supplementary theorem for parallel lines cut by a transversal.)