Question

The rails of a railroad track must always be parallel. Suppose ∠1 and ∠2 have the same measures. What must be true about their measures to ensure the rails are parallel?

a. ∠1 must be supplementary to ∠2 because it is a corresponding angle of ∠2.
b. ∠1 must be a right angle because it is an alternate interior angle of ∠2.
c. ∠1 and ∠2 must be right angles because they are congruent and supplementary consecutive interior angles.
d. ∠1 is a consecutive interior angle of ∠2, but its measure cannot be determined.

Answer 1

To ensure the railroad tracks are parallel, ∠1 and ∠2 must be both right angles, as they are congruent and supplementary consecutive interior angles.

Step-by-step explanation:

To ensure that the rails of a railroad track are parallel, the angles mentioned must have certain relationships. Specifically, if ∠1 and ∠2 are the same measures and we want the rails to be parallel, option c from the given choices would be the correct answer. This is because congruent and supplementary consecutive interior angles together would imply that both angles add up to 180 degrees, thus making each angle a right angle, or 90 degrees. Having right angles ensures the tracks are parallel.

As per mathematical principles, parallel lines cut by a transversal line create pairs of angles that are either congruent or supplementary. For instance, alternate interior angles or corresponding angles are congruent, and consecutive interior angles are supplementary. If the consecutive interior angles are also congruent, as stated in the question, the only way for them to be supplementary and congruent is if they are both right angles.