Question

given that lines a and b are parallel, angles 1 and 5 are congruent because they are corresponding angles, and angles 1 and 4 are congruent because they are vertical angles. then, by which property are angles 4 and 5 congruent? together, these statements prove which theorem? a transitive property of congruency; if parallel lines have a transversal, then supplemental angles are congruent. b transitive property of congruency;if parallel lines have a transversal, then alternate interior angles are congruent. c symmetric property of congruency; if parallel lines have a transversal, then alternate exterior angles are congruent. d commutative property of congruency; if parallel lines have a transversal, then same side interior angles are congruent.

Explanation:

• Transitive Property of Congruency: If \( \angle1\cong\angle5\) (corresponding angles) and \( \angle1\cong\angle4\) (vertical angles), then by the transitive property \( \angle4\cong\angle5\) (\( \angle1\cong\angle5\) and \( \angle1\cong\angle4\) implies \( \angle4\cong\angle5\)).
• Alternate - Interior Angles Theorem: Angles \(4\) and \(5\) are alternate - interior angles. The transitive property is used to show their congruency. The theorem “If parallel lines have a transversal, then alternate interior angles are congruent” is proved. The symmetric property states that if \( \angle A\cong\angle B\), then \( \angle B\cong\angle A\). The commutative property is for operations (not relevant here). And supplementary angles are not congruent (they add up to \(180^{\circ}\) unless they are right angles).

Answer:

B. Transitive Property of Congruency; If parallel lines have a transversal, then alternate interior angles are congruent.