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:

Given \( m \parallel \overline{CB} \), the transversal \( \overline{AB} \) intersects these parallel lines. \( \angle XAB \) and \( \angle ABC \) lie on opposite sides of the transversal and between the parallel lines, so they are alternate interior angles. The theorem stating alternate interior angles of parallel lines cut by a transversal are congruent justifies their congruence.

Since line \( m \parallel \overline{CB} \) and transversal \( \overline{AB} \) crosses them, \( \angle XAB \) and \( \angle ABC \) are alternate interior angles. The theorem for alternate interior angles of parallel lines confirms their congruence.

Line \( m \parallel \overline{CB} \), transversal \( AB \) creates alternate interior angles \( \angle XAB \) and \( \angle ABC \), which are congruent by the alternate interior angles theorem.

Answer:

A. Corresponding angles of parallel lines cut by a transversal are congruent. (Note: Correction—actual correct option is the one about alternate interior angles; assuming the last option is D: Alternate interior angles of parallel lines cut by a transversal are congruent. So correct answer is D. Alternate interior angles of parallel lines cut by a transversal are congruent.)

Wait, rechecking: The options list the last one as "Alternate interior angles of parallel lines cut by a transversal are congruent." So that's the correct justification. So final answer: