Question

consider the diagram. given that ( r parallel s ) and ( q ) is a transversal, we know that ( angle 3 cong angle 6 ) by the ______. ( \bigcirc ) corresponding angles theorem ( \bigcirc ) alternate interior angles theorem ( \bigcirc ) vertical angles theorem ( \bigcirc ) alternate exterior angles theorem

Explanation:

To determine the theorem for \(\angle 3 \cong \angle 6\) when \(r \parallel s\) and \(q\) is a transversal:

• Corresponding Angles Theorem: Applies to angles in the same position relative to parallel lines and transversal (e.g., \(\angle 1\) and \(\angle 5\)), not \(\angle 3\) and \(\angle 6\).
• Alternate Interior Angles Theorem: States that if two parallel lines are cut by a transversal, alternate interior angles (between the lines, opposite sides of transversal) are congruent. \(\angle 3\) and \(\angle 6\) are between \(r\) and \(s\) (interior) and on opposite sides of \(q\) (alternate), so this theorem applies.
• Vertical Angles Theorem: Applies to angles opposite each other when two lines intersect (e.g., \(\angle 3\) and \(\angle 2\)), not here.
• Alternate Exterior Angles Theorem: Applies to angles outside the parallel lines (e.g., \(\angle 1\) and \(\angle 8\)), not \(\angle 3\) and \(\angle 6\).

Answer:

B. alternate interior angles theorem