Question

If Angle 2 is congruent to angle 4 and Angle 5 is congruent to angle 7, which describes all the lines that must be parallel?

Lines r and s are crossed by lines t and u to form 16 angles. Clockwise from top left, at the intersection of r and t, the angles are 1, 2, 3, 4; at the intersection of s and t, 5, 6, 7, 8; at the intersection of u and s, 9, 10, 11, 12; at the intersection of u and r, 13, 14, 15, 16.
Only lines r and s must be parallel.
Only lines t and u must be parallel.
Lines r and s and lines t and u must be parallel.
Neither lines r and s nor lines t and u must be parallel.

Answer 1

Neither lines r and s nor lines t and u must be parallel.

Reason: Angle 2 and angle 4 are vertical angles at the same intersection (r ∩ t), and angle 5 and angle 7 are vertical angles at the same intersection (s ∩ t). Vertical angles are always congruent regardless of whether any of the lines are parallel, so no parallelism is forced.