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 position relative to the parallel lines and transversal. But \(\angle XAB\) and \(\angle ABC\) are not in corresponding positions.
• Vertical angles: Vertical angles are formed by two intersecting lines. Here, there is no such intersection to form vertical angles between \(\angle XAB\) and \(\angle ABC\).
• Same - side interior angles: These angles add up to \(180^{\circ}\) (supplementary). But we want congruence (\(=\) measure), not supplementary (\(= 180^{\circ}\) sum).
• Alternate interior angles: When two parallel lines (\(m\parallel CB\)) are cut by a transversal (\(AB\)), alternate interior angles (\(\angle XAB\) and \(\angle ABC\)) are congruent.

Answer:

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