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Use the figure and flowchart proof to answer the question:

Segments UV and WZ are parallel segments that intersect line ST at points Q and R, respectively

Points S, Q, R, and T all lie on the same line; Given. Arrows are drawn from this statement to the following three statements. Statement 1: The measure of angle SQT equals 180 degrees; Reason 1: Definition of a Straight Angle. Statement 2: The measure of angle SQV plus the measure of angle VQT equals the measure of angle SQT; Reason 2: Angle Addition Postulate. Statement 3: The measure of angle SQV plus the measure of angle VQT equals 180 degrees; Reason 3: Substitution Property of Equality. Lines UV and WZ are parallel; Given. An arrow is drawn from this statement to the following statements. Statement 4: The measure of angle VQT plus the measure of angle ZRS equals 180 degrees; Reason A. Statement 5: The measure of angle SQV plus the measure of angle VQT equals the measure of angle VQT plus the measure of angle ZRS; Reason B. An arrow also points from Statement 3 to Statement 5. An arrow from Statement 5 points to the following statements. Statement 6: The measure of angle SQV plus the measure of angle VQT minus the measure of angle VQT equals the measure of angle VQT plus the measure of angle ZRS minus the measure of angle VQT, the measure of angle SQV equals the measure of angle ZRS; Reason C. Statement 7: The measure of angle SQV is congruent to the measure of angle ZRS; Definition of Congruency.

Which theorem accurately completes Reason A?

Alternate Interior Angles Theorem
Corresponding Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Interior Angles Theorem

1 answer

To determine which theorem accurately completes Reason A ("The measure of angle VQT plus the measure of angle ZRS equals 180 degrees"), we need to analyze the situation with respect to the properties of parallel lines and the angles formed when a transversal intersects those lines.

Given that segments UV and WZ are parallel and intersected by line ST at points Q and R, angles VQT and ZRS are formed.

For angles formed in this configuration, we need the relationship between angles:

  1. Alternate Interior Angles Theorem: States that if two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent.

  2. Corresponding Angles Theorem: States that if two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent.

  3. Alternate Exterior Angles Theorem: States that if two parallel lines are cut by a transversal, then each pair of alternate exterior angles are congruent.

  4. Same-Side Interior Angles Theorem: States that if two parallel lines are cut by a transversal, then the same-side interior angles are supplementary (their measures add up to 180 degrees).

Given the angles VQT and ZRS lie on the same side of the transversal ST, and both angles are located inside the parallel lines UV and WZ, they are classified as same-side interior angles.

Therefore, the correct theorem that applies here and completes Reason A is the Same-Side Interior Angles Theorem.