Points, Lines and Angles

Lessons

Senior 1 · Mathematics

Points, Lines and Angles

5 lessons

Lesson 4 of 5

Parallel lines and a transversal

25 min

Stripes on the road

The white stripes of a zebra crossing are painted parallel to each other, and the edge of the road runs across all of them. Because the stripes are parallel, the edge meets every stripe at the same angle. That observation is the heart of this lesson: when a line crosses two parallel lines, the angles it makes come in pairs, and the pairs are either equal or add up to $180^\circ$ .

Definition

Transversal: a line that crosses two or more other lines at different points.

Example: in the figure below, $t$ is a transversal of the parallel lines $l$ and $m$ . It makes four angles at $P$ and four at $Q$ .

$l \parallel m$ , and the transversal $t$ makes eight angles. Here $a = c = e = g = 60^\circ$ and $b = d = f = h = 120^\circ$ .

The two crossings look exactly alike, so it helps to name the angles by position. The angles between the two lines ( $c, d, e, f$ ) are interior angles; the others ( $a, b, g, h$ ) are exterior angles.

Definition

Corresponding angles: two angles in the same position at the two crossings, for example both above their line and to the right of the transversal.

Example: $a$ and $e$ , $b$ and $f$ , $c$ and $g$ , $d$ and $h$ .

Definition

Alternate angles: two interior angles on opposite sides of the transversal, one at each crossing.

Example: $c$ and $e$ ; $d$ and $f$ .

Definition

Co-interior angles (also called allied or same-side interior angles): two interior angles on the same side of the transversal, one at each crossing.

Example: $c$ and $f$ ; $d$ and $e$ .

Trace the letter to find the pair. The letters may be turned round, reflected or stretched.

Important

When a transversal crosses two PARALLEL lines

Corresponding angles are equal ( $a = e$ ). Parallel lines point in the same direction, so the transversal makes the same turn onto each of them. This is the basic fact of parallel lines; the other two rules follow from it.
Alternate angles are equal ( $c = e$ ). Why: $c = a$ (vertically opposite angles) and $a = e$ (corresponding angles), so $c = e$ .
Co-interior angles add up to $180^\circ$ ( $c + f = 180^\circ$ ). Why: $e + f = 180^\circ$ (angles on a straight line) and $c = e$ (alternate angles), so $c + f = 180^\circ$ .

Important

The test for parallel lines. The rules also work backwards. Two lines cut by a transversal are parallel if any one of these is true:

a pair of corresponding angles are equal;
a pair of alternate angles are equal;
a pair of co-interior angles add up to $180^\circ$ .

Worked example 1: all eight angles

One angle is given; find the other seven.

The two lines are parallel. Find $b, c, d, e, f, g$ and $h$ , giving reasons.

Solution

$b = 180^\circ - 70^\circ = 110^\circ$ (angles on a straight line)
$c = 70^\circ$ (vertically opposite to the $70^\circ$ angle)
$d = b = 110^\circ$ (vertically opposite angles)
$e = 70^\circ$ (corresponding to the $70^\circ$ angle)
$f = b = 110^\circ$ (corresponding angles)
$g = c = 70^\circ$ (corresponding angles)
$h = d = 110^\circ$ (corresponding angles)

Once one angle is known, every angle is either that angle or its supplement.

Worked example 2: an equation from co-interior angles

Two co-interior angles.

The lines are parallel. Find $x$ and both angles.

Solution The two angles lie between the lines, on the same side of the transversal, so they are co-interior:

(3x - 5) + (2x + 10) = 180 \quad \text{(co-interior angles)}

5x + 5 = 180, \quad 5x = 175, \quad x = 35.

The angles are $3(35) - 5 = 100^\circ$ and $2(35) + 10 = 80^\circ$ . Check: $100^\circ + 80^\circ = 180^\circ$ . ✓

Worked example 3: are the lines parallel?

Are l and m parallel? They look it.

Solution The $106^\circ$ and $72^\circ$ angles are co-interior. If $l \parallel m$ , they would add up to $180^\circ$ . But

106^\circ + 72^\circ = 178^\circ \ne 180^\circ,

so $l$ and $m$ are not parallel. On the right of the transversal the co-interior angles add up to less than $180^\circ$ , so the lines slowly close in and meet far away on that side. Two degrees cannot be seen by eye, which is why geometry decides with angles, not with how a drawing looks.

Worked example 4: the angle between two parallel lines

The dashed line through $E$ is parallel to $A B$ and $C D$ . It splits $\angle BED$ into two alternate angles.

$AB \parallel CD$ , $\angle ABE = 35^\circ$ and $\angle CDE = 50^\circ$ . Find $\angle BED$ .

Solution No rule applies directly, because $E$ is not on either line. So draw a construction line through $E$ , parallel to $A B$ and $C D$ . It splits $\angle BED$ into two parts:

the upper part equals $\angle ABE = 35^\circ$ (alternate angles, with $B E$ as the transversal);
the lower part equals $\angle CDE = 50^\circ$ (alternate angles, with $D E$ as the transversal).

\angle BED = 35^\circ + 50^\circ = 85^\circ.

Warning

Check for the arrows first

The F, Z and C rules hold only when the lines are parallel. If a diagram has no arrow marks and the question does not say the lines are parallel, you may not assume it (see Worked example 3).
The letters are often turned round: a Z may appear as an S, an F upside down or back to front, and a C as a U. Look for the position of the angles, not for the letter drawn upright.
Co-interior angles are not equal (unless both are $90^\circ$ ): they add up to $180^\circ$ .

Tip

F for corresponding, Z for alternate, C for co-interior. F and Z angles are equal; C angles are supplementary. A quick check: if one angle of the pair is acute and the other obtuse, they cannot be F or Z angles.

Cognito: Alternate, Corresponding and Allied Angles (6 min). "Allied angles" is another name for co-interior angles. Watch how each pair is found by its position and how the reason is written after each angle.

✦

Practice

Questions 1 to 3 use the figure of the eight angles $a$ to $h$ above ( $l \parallel m$ ).

Name (a) the angle corresponding to $d$ , (b) the angle alternate to $c$ , (c) the angle co-interior with $d$ .
If $a = 72^\circ$ , find $e$ , $c$ and $f$ , giving a reason for each.
True or false? (a) $b = h$ (b) $c + h = 180^\circ$ (c) $d = e$ .
Two co-interior angles between parallel lines are $3 y$ and $2 y$ . Find $y$ and both angles.
Two alternate angles between parallel lines are $(4x - 20)^\circ$ and $(2x + 30)^\circ$ . Find $x$ and the angles.
A transversal crosses two lines. A pair of corresponding angles measure $83^\circ$ and $85^\circ$ . Are the lines parallel? Explain.
In a figure like Worked example 4, $AB \parallel CD$ , $\angle ABE = 42^\circ$ and $\angle CDE = 57^\circ$ . Find $\angle BED$ .

Show answersHide answers

(a) $h$ (b) $e$ (c) $e$
$e = 72^\circ$ (corresponding to $a$ ); $c = 72^\circ$ (vertically opposite to $a$ ); $f = 108^\circ$ (co-interior with $c$ ).
(a) True: $b = f$ (corresponding angles) and $f = h$ (vertically opposite angles). (b) True: $c = g$ (corresponding angles) and $g + h = 180^\circ$ (angles on a straight line). (c) False: $d$ and $e$ are co-interior, so $d + e = 180^\circ$ ; they are equal only if both are $90^\circ$ .
$5y = 180^\circ$ , so $y = 36^\circ$ ; the angles are $108^\circ$ and $72^\circ$ .
$4 x - 20 = 2 x + 30$ , so $x = 25$ ; both angles are $80^\circ$ .
No. If the lines were parallel, corresponding angles would be equal; $83^\circ \ne 85^\circ$ .
$42^\circ + 57^\circ = 99^\circ$ (draw a construction line through $E$ and use alternate angles twice).

Sources and credits

10.2: Angles (transversals and parallel lines), from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Photo: Zebra Crossing Kigali.jpg, Mrbobax, Wikimedia Commons, licensed under CC BY-SA 4.0.
Video: GCSE Maths - Alternate, Corresponding and Allied Angles, Cognito, embedded from YouTube.

The proofs, the diagrams, the worked examples, the practice items and the quiz questions were written by Ganzaa.

1Stripes on the road

Stripes on the road