Question

given: m || cb
prove: m∠abc + m∠bac + m∠acb = 180°
which statement justifies that angle xab is congruent to angle abc?
○ corresponding angles of parallel lines cut by a transversal are congruent.
○ vertical angles are congruent.
○ same - side interior angles of parallel lines cut by a transversal are supplementary.
○ alternate interior angles of parallel lines cut by a transversal are congruent.

Explanation:

To determine which statement justifies that ∠XAB ≅ ∠ABC, we analyze the parallel lines \( m \parallel \overline{CB} \) and the transversal \( \overline{AB} \).

• Corresponding angles: Require positions like "same - side, same - relative - position" which doesn't fit ∠XAB and ∠ABC.
• Vertical angles: Are formed by intersecting lines, not related to parallel lines cut by a transversal here.
• Same - side interior angles: Are supplementary, not congruent in general for parallel lines cut by a transversal, so this is incorrect.
• Alternate interior angles: When parallel lines are cut by a transversal, alternate interior angles are congruent. ∠XAB and ∠ABC are alternate interior angles formed by transversal \( \overline{AB} \) cutting \( m \) and \( \overline{CB} \), so this statement justifies their congruence.

Answer:

D. Alternate interior angles of parallel lines cut by a transversal are congruent.