Theorem
Angles made by a transversal of two parallel lines
Geometry • Line • Parallel Lines and Transversals
Statement
When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior (allied) angles on the same side of the transversal are supplementary; each converse also holds, so any one of those relations proves the two lines parallel.
How to use it
- 1Carry a known angle across to the other parallel line as its equal corresponding or alternate angle.
- 2Use the co-interior pair whenever two angles at the two ends of one side of a parallelogram or trapezium are wanted: they add to .
- 3Run it backwards: a pair of equal alternate angles is enough to conclude that the two lines are parallel.
Worked example
A transversal cuts two parallel lines and one of the eight angles formed is . What are its co-interior partner and its alternate interior angle?
Step-by-step
- 1co-interior:
Why: Co-interior angles on the same side of the transversal are supplementary.
- 2alternate interior:
Why: Alternate interior angles are equal.
Answer
and
Watch out
Every one of these relations needs the two lines to be parallel; across non-parallel lines a transversal gives no equality at all. Do not confuse them with vertically opposite angles, which are equal at any crossing whatsoever.
Builds on
Also known as
corresponding angles, alternate angles, co-interior angles, allied angles, transversal of parallel lines, F angles Z angles C angles