Question

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 angles 5 and 7 are congruent because vertical angles cut by a transversal are congruent. therefore, angles 2 and 7 are congruent ba

Explanation:

Step1: Recall angle - pair relationships

When two parallel lines are cut by a transversal, corresponding angles are congruent. In the given figure, lines \(a\) and \(b\) are parallel and \(c\) is a transversal. Angle 2 and angle 5 are corresponding angles, so they are congruent because of the corresponding - angles postulate.

Step2: Recall vertical - angle property

Vertical angles are always congruent. Angles 5 and 7 are vertical angles. So, angle 5 and angle 7 are congruent because vertical angles are congruent.

Step3: Use the transitive property of congruence

Since \(\angle2\cong\angle5\) and \(\angle5\cong\angle7\), by the transitive property of congruence, \(\angle2\cong\angle7\).

Answer:

Angles 2 and 5 are congruent because corresponding. Angles 5 and 7 are congruent because vertical. Therefore, angles 2 and 7 are congruent by the transitive property of congruence.