Question

given: m || \overline{cb}
prove: m\angle abc + m\angle bac + m\angle acb = 180^circ
which statement justifies that angle xab is congruent to angle abc?
\bigcirc corresponding angles of parallel lines cut by a transversal are congruent.
\bigcirc vertical angles are congruent.
\bigcirc same - side interior angles of parallel lines cut by a transversal are supplementary.
\bigcirc 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 \( \overline{AB} \). ∠XAB and ∠ABC are alternate interior angles formed by the transversal \( \overline{AB} \) intersecting the parallel lines \( m \) and \( \overline{CB} \). The theorem "Alternate interior angles of parallel lines cut by a transversal are congruent" applies here. Corresponding angles would involve a different position, vertical angles are formed by intersecting lines (not parallel lines with a transversal in this case), and same - side interior angles are supplementary, not congruent for this pair.

Answer:

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