Parallel Lines and a Transversal – Alternate and Corresponding Angles

What it shows

A transversal c crossing lines a and b forms eight angles, four at A and four at B. Vertically opposite angles are always equal, whatever the lines do. When a ∥ b, alternate interior angles are equal, corresponding angles are equal and co-interior angles add up to 180°. The converse gives a test for parallel lines: if one pair of alternate interior or corresponding angles is equal, or one pair of co-interior angles is supplementary, then a ∥ b. Tilting b by even a few degrees breaks all of these relations at once, so one figure shows both the property and its converse.

How to use

Choose an Angle pair to highlight, for example Alternate interior, and start with b parallel to a. Change the Angle between transversal c and line a and note that the highlighted pair stays equal. Then drag the end of line b, or change the Tilt of b relative to a, and read how far the pair is from being equal. Press Set b ∥ a to restore the parallel position, and repeat with Corresponding and Co-interior.

Parameters you can change

• Angle between transversal c and line a 20–160 °
• Tilt of b relative to a (0 = parallel) -30–30 °
• Angle pair to highlight Alternate interior angles, Corresponding angles, Co-interior angles, All 8 angles
• Show angle names (A1…B4)

Questions to explore

1. If two co-interior angles add up to 180°, what can you conclude about lines a and b?
2. When a is not parallel to b, which pairs of angles are still always equal?
3. If b is tilted by 10° from the parallel position, by how much do the alternate interior angles differ?