📐 Geometry
🛤️Parallel Lines & Transversals
A third line slicing through two parallels creates eight angles of only two sizes — the F, Z and C shapes tell you which is which.
Railway tracks, zebra crossings, the ruled lines of a notebook — alongside angles, parallel lines are geometry's most familiar neighbour. Two parallel lines live in peace until a third line cuts across them: then eight angles step on stage at once, and the story gets busy.
Parallel lines and a transversal
Parallel lines are two lines in the same plane that never meet; diagrams mark them with arrows — one arrow each for a pair, two arrows each for another pair.
The third line cutting across both is a transversal. One slice, two crossing points, 8 angles born at once. It looks dizzying, but the whole picture holds only two sizes of angle — and three relationships string them all together.
Corresponding angles: the F shape
Two angles on the same side of the transversal, each occupying the same position at its own crossing, are corresponding angles. Trace the path and you draw an F.
Corresponding angles are equal: if the transversal meets the first line at , the angle in the same position on the second line is too.
Alternate angles Z, co-interior angles C
Angles sitting between the two parallels on opposite sides of the transversal are alternate angles — trace them and you draw a Z. They are equal as well: 65° meets 65°.
Angles between the two lines but on the same side of the transversal are co-interior angles, shaped like a C. They are not equal, but they sum to 180°: , so the partner of 65° is .
One angle lights up all eight
Given a single 65° angle, chase out the other seven: it forms a straight line with its neighbour, giving ; its vertically opposite angle is 65° again. Crossing the transversal to the second parallel: the corresponding angle is 65°, the alternate angle is 65°, the co-interior angle is 115°. Count them all — the eight angles wear only two faces, 65° and 115°.
Why equal: slide the angle along
Imagine sliding the 65° angle along the transversal to the other crossing point: during the slide, neither arm changes direction, so the opening cannot change either — that is exactly why corresponding angles are equal. The equality of alternate angles and the supplement of co-interior angles both follow from this sliding angle.
So all three relationships grow from one root: parallel lines let angles travel freely along the transversal. This trick goes on doing heavy work when you hunt interior angle sums in angles of polygons.
180° = π rad
Dial up a 65° acute angle and memorise its look; then dial its corresponding angle — the size does not budge. Try 115°: the slightly obtuse twin that fits together with 65° into one straight line.
Check yourself
Quick quiz
1. Two parallels are cut by a transversal; one corresponding angle measures 72°. What is the other corresponding angle?
2. What relation holds between a pair of co-interior angles?
3. One alternate angle measures 65°. What is its alternate partner?