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 supplementary 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: The transitive property states that if \(a = b\) and \(b = c\), then \(a = c\). In this case, \(\angle1\cong\angle5\) (corresponding angles) and \(\angle1\cong\angle4\) (vertical angles). By the transitive property of congruency, if \(\angle4\cong\angle1\) and \(\angle1\cong\angle5\), then \(\angle4\cong\angle5\). This helps in proving the theorem related to parallel - line angle congruency.

Answer:

A. Transitive Property of Congruency: If parallel lines have a transversal, then supplementary angles are congruent.