Question

given: a and b are parallel and c is a transversal.
prove: \\( \angle 2 \cong \angle 7 \\)
use the drop - down menus to complete the paragraph proof showing that alternate interior angles are congruent.
we know that lines a and b are parallel and that line c is a transversal because that is given. we can tell that angles 2 and 5 are congruent because \\( \square \\) angles are congruent. angles 5 and 7 are congruent because \\( \square \\) angles by parallel lines cut by a transversal are congruent. therefore, angles 2 and 7 are congruent based on the \\( \square \\)

Explanation:

Step1: Vertical angles

Vertical angles are formed when two lines intersect. By the vertical angles theorem, vertical angles are congruent. So, \(\angle2\) and \(\angle5\) are vertical angles.

Step2: Alternate exterior

When a transversal intersects two parallel lines, alternate exterior angles are congruent. Here, \(\angle5\) and \(\angle7\) are alternate exterior angles with respect to parallel lines \(a\) and \(b\) and transversal \(c\).

Step3: Transitive property

The transitive property of congruence states that if \(\angle A\cong\angle B\) and \(\angle B\cong\angle C\), then \(\angle A\cong\angle C\). Since \(\angle2\cong\angle5\) and \(\angle5\cong\angle7\), by the transitive property, \(\angle2\cong\angle7\).

Answer:

First blank: vertical; Second blank: alternate exterior; Third blank: transitive property of congruence.