Question

Which two statements, if true, guarantee that line RV is parallel to line PQ?

1) The circle arc with center at point t passes through point s
2) The circle arc with center at point r intersects ray PR at point u
3) The line passing through points r and v is perpendicular to line PQ
4) The circle arc with center at point u intersects arc R at point v

Answer 1

To determine if line RV is parallel to line PQ, we must examine the relationship of angles and lines. Statement (3) concerning a line perpendicular to PQ confirms parallelism, while others do not directly relate to parallel lines.

Step-by-step explanation:

To determine which two statements guarantee that line RV is parallel to line PQ, we need to look for statements that relate to the angle relationships or the properties of parallel lines. Statements that demonstrate congruent corresponding angles, alternate interior angles, or the perpendicularity of transversal lines would be strong indicators of parallel lines.

Statement (3) "The line passing through points r and v is perpendicular to line PQ" would mean that RV creates corresponding angles with any line that is also perpendicular to PQ, thus RV would be parallel to PQ under these conditions.

None of the other statements (1, 2, or 4) establish a direct relationship of angles or geometry that relates RV to PQ in a way that would necessitate parallelism. Considering the principles of geometry, especially the Corresponding Angle Postulate and the Perpendicular Transversal Theorem, it is evident that only statement (3) directly guarantees RV is parallel to PQ.