Question

given: ( m parallel overline{cb} ) prove: ( mangle abc + mangle bac + mangle acb = 180^{circ} ) 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:

• Corresponding angles: These are in the same relative position. \( \angle XAB\) and \( \angle ABC\) are not in corresponding positions.
• Vertical angles: These are opposite angles formed by two intersecting lines. \( \angle XAB\) and \( \angle ABC\) are not formed by two intersecting lines.
• Same - side interior angles: These add up to \(180^{\circ}\). But we want congruence (equality in measure), not supplementary (sum to \(180^{\circ}\)).
• Alternate interior angles: When two parallel lines (\(m\) and \(CB\)) are cut by a transversal (\(AB\)), alternate interior angles are congruent. \( \angle XAB\) and \( \angle ABC\) are alternate interior angles with respect to parallel lines \(m\) and \(CB\) and transversal \(AB\).

Answer:

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