A line across two parallel lines
A line that crosses two or more other lines is called a transversal. When a transversal crosses two parallel lines, it makes two crossings, and each crossing has four angles around it: eight angles in all.
Name the four at the upper crossing a, b, c and d, and the four at the lower crossing e, f, g and h, in the same positions: upper left, upper right, lower left, lower right. The arrowheads on the two lines say that they are parallel.
The transversal crosses the two parallel lines and makes eight angles, four at each crossing.
Corresponding angles are equal
The two parallel lines point in exactly the same direction, so the transversal crosses both of them at the same slant. Slide the upper crossing down the transversal onto the lower one. The upper line never turns as it slides, so it lands exactly on the lower line, and each of the four upper angles lands on the angle in the same position below it.
Angles in matching positions at the two crossings are called corresponding angles, and they are equal: a = e, b = f, c = g and d = h. A corresponding pair often makes the shape of a letter F.
the eight angles are only two sizes, θ and 180° − θ, four of each: alternate, corresponding and co-interior are three names for that
Slide the upper crossing down onto the lower one
The transversal crosses the parallel lines at . Drag the handle to the left of the upper crossing downward: the upper crossing slides along the transversal, and its four angles land exactly on the four at the lower crossing. The handle at the upper crossing turns the transversal, and the angles still match at every slant.
Alternate angles are equal
Alternate angles lie between the two parallel lines, on opposite sides of the transversal, such as c and f. Together with the transversal they make the shape of a letter Z.
They are equal, and two earlier facts show why. c and b are vertically opposite at the upper crossing, so c = b. b and f are corresponding angles, so b = f. So c = f. In the same way, d = e.
Alternate angles, between the parallel lines on opposite sides of the transversal: both are 70°.
Co-interior angles add up to 180°
Co-interior angles lie between the two parallel lines, on the same side of the transversal, such as c and e. Together they make the shape of a letter C.
They are not equal: they add up to 180°. e and f sit side by side on the lower parallel line, so e + f = 180. f = c, because they are alternate angles. So c + e = 180. If c is 70°, then e is 180 − 70 = 110°.
Co-interior angles, between the parallel lines on the same side of the transversal: 70 + 110 = 180.
Only two sizes
Put the three facts together, and all eight angles come in only two sizes. If one angle is 65°, each of the eight is either 65° or 180 − 65 = 115°, four of each. The acute ones are all equal, the obtuse ones are all equal, and an acute angle and an obtuse angle always add up to 180°.
The usual mistakes: making every angle equal, when only the four of one size are; using 180 − a for corresponding or alternate angles, which are equal; and using a for co-interior angles, which add up to 180°.
The rules need the lines to be parallel. If a diagram has no arrow marks and nothing says the lines are parallel, angles that look equal prove nothing, because two lines that are slightly tilted toward each other make angles of different sizes.
Worked example: Parallel Lines with One Transversal
Question AB is parallel to CD. A straight line EF cuts AB at G and CD at H. ∠ EGB = 118°. Find ∠ GHD and ∠ CHG.
1.AB ∥ CD and EF crosses both, so each angle at G has an equal partner at H.
EF crosses the parallels AB and CD at G and H. One angle is given. 2.∠ EGB and ∠ GHD are corresponding angles (the F shape): ∠ GHD = 118°.
F shape: ∠ EGB and ∠ GHD are corresponding, so ∠ GHD = 118°. 3.CD is a straight line, so ∠ CHG = 180° − 118° = 62°.
On the straight line CD: ∠ CHG = 180° − 118° = 62°. 4.Check with alternate angles (the Z shape): ∠ BGH = 180° − 118° = 62°, and ∠ BGH = ∠ CHG.
Z shape: ∠ BGH = 62°, alternate to ∠ CHG. 5.Check with co-interior angles (the C shape): ∠ BGH + ∠ GHD = 62° + 118° = 180°.
C shape: ∠ BGH + ∠ GHD = 62° + 118° = 180°.
Answer: ∠ GHD = 118°; ∠ CHG = 62°
Common mistakes
- Making every angle at the transversal 118°: only the four obtuse ones are; the other four are 62°.
- Calling ∠ EGB and ∠ CHG alternate angles: they are on the same side of the transversal, and are neither equal nor a Z pair.
Worked example: Parallel Lines with a Bent Transversal
Question AB is parallel to CD. Point E lies between the two lines, with ∠ ABE = 34° and ∠ EDC = 51°. Find ∠ BED.
1.E sits between the parallel lines, so no single Z or F shape reaches from B to D.
AB and CD are parallel. The path B-E-D bends at E, between them. 2.Draw a line through E parallel to AB and CD, and mark a point X on it to the right of E.
A line through E, parallel to both, is drawn; X is on it, to the right. 3.AB ∥ EX: ∠ BEX is alternate to ∠ ABE, so ∠ BEX = 34°.
Z shape with AB: ∠ BEX = ∠ ABE = 34°. 4.CD ∥ EX: ∠ XED is alternate to ∠ EDC, so ∠ XED = 51°.
Z shape with CD: ∠ XED = ∠ EDC = 51°. 5.∠ BED is the two parts together: 34° + 51° = 85°.
∠ BED = 34° + 51° = 85°. 6.The line through E is the whole method: it turns one bent transversal into two straight ones.
Answer: ∠ BED = 85°
Common mistakes
- Subtracting, 51° − 34° = 17°: the parallel through E shows the two alternate angles sit side by side, so they add.
- Answering 180° − 85° = 95° from an imagined triangle BED; BD is not drawn and nothing says it is.