Angles in Parallel Lines

Use corresponding, alternate and co-interior angles on parallel lines, with the exact reasons, to find angles, solve for x and prove lines are parallel.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 58 real past-paper questions below

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Three Pairs of Angles

Parallel lines never meet. They are marked with matching arrows. A line that crosses them is called a transversal.

Where the transversal crosses each line, it makes the same four angles. So angles pair up.

Pair	Reason to write
F shape: equal	corresponding angles are equal
Z shape: equal	alternate angles are equal
C shape: add to 180°	co-interior angles add up to 180°

Worked example

1
$one angle = 6 5^{\circ}$ — given
2
$corresponding = 6 5^{\circ}$ — corresponding angles are equal
3
$co-interior = 180 - 65 = 11 5^{\circ}$ — co-interior angles add up to 180°

Common mistake

Writing "Z angles" or "F angles" as the reason. The exam wants the words alternate, corresponding, co-interior.

Key idea

Around 240 BC, Eratosthenes in Alexandria, Egypt, measured the Earth with parallel lines. The Sun's rays arrive almost parallel. At noon the Sun was overhead in Syene (Aswan) but about 7.2° from vertical in Alexandria. By alternate angles, the cities are 7.2° apart: $\frac{1}{50}$ of the way round the Earth.

One Angle, Then a Triangle

The 4 or 5-mark question: jump across the parallel lines, then finish inside a triangle. Give a reason for each stage.

$P Q$ is parallel to $R S$ . $E$ is on $P Q$ , $B$ and $C$ are on $R S$ , and $E B = E C$ . Angle $P E B = 5 2^{\circ}$ . Find angle $B E C$ .

Worked example

1
$∠ E B C = 5 2^{\circ}$ — Alternate angles are equal: angle PEB and angle EBC make a Z between PQ and RS.
2
$∠ E C B = 5 2^{\circ}$ — EB = EC, so the triangle is isosceles. Its two base angles are equal.
3
$∠ B E C = 180 - 52 - 52$ — Angles in a triangle add up to 180°, so take both base angles from 180.
4
$∠ B E C = 7 6^{\circ}$ — 180 − 104 = 76. This is the angle the question asks for.

Common mistake

Giving the number with no reason. Each line needs its own reason to get the marks.

Two Lines Cross the Parallels

A very common question: two straight lines cross a pair of parallel lines and meet on one of them. They make a triangle. Use a Z, a straight line, then the triangle.

$A B$ is parallel to $C D$ . Two straight lines meet at $P$ on $A B$ and cross $C D$ at $Q$ and $R$ , making triangle $P QR$ . Angle $A P Q = 6 4^{\circ}$ . Outside the triangle, on the straight line $QR D$ , angle $P R D = 12 1^{\circ}$ . Find angle $QP R$ .

Worked example

1
$∠ P QR = 6 4^{\circ}$ — Alternate angles are equal: angle APQ and angle PQR make a Z between the parallel lines.
2
$∠ P R Q = 180 - 121 = 5 9^{\circ}$ — Angles on a straight line add up to 180°: angle PRQ and the 121° sit together on line QRD.
3
$∠ QP R = 180 - 64 - 59$ — Angles in a triangle add up to 180°, so take the two angles you know from 180.
4
$∠ QP R = 5 7^{\circ}$ — 180 − 123 = 57. This is the angle the question asks for.
5
$64 + 57 + 59 = 180$ — Check: at P, the three angles along line AB make a straight line. Angle BPR is 59°, alternate with angle PRQ.

Common mistake

Putting the 121° straight into the triangle. It is outside, on the straight line, so find 180 − 121 first.

Key idea

Write each angle on the diagram as soon as you find it. Examiners give a mark for a correct angle in the right place, even if the final answer goes wrong.

Two Pairs of Parallel Lines

Some diagrams have two pairs of parallel lines. Jump across one pair at a time, and say which pair you used.

$A B$ is parallel to $D C$ , and $A D$ is parallel to $B C$ . Together they make the shape $A B C D$ . Angle $D A B = 6 8^{\circ}$ . Find angle $A B C$ and angle $B C D$ .

Worked example

1
$∠ A B C = 180 - 68 = 11 2^{\circ}$ — AD is parallel to BC, and AB crosses both. Angles DAB and ABC are co-interior, so they add up to 180°.
2
$∠ B C D = 180 - 112 = 6 8^{\circ}$ — Now the other pair: AB is parallel to DC, and BC crosses both. Angles ABC and BCD are co-interior.
3
$68 + 112 + 68 + 112 = 360$ — Check: the four angles of the shape add up to 360°, as in any four-sided shape.

Key idea

Some books call co-interior angles allied or interior angles. All three names mean the same pair, and all three are accepted.

Common mistake

Using the wrong pair of lines. Before each step, find the two parallel lines and the line that crosses them.

Co-interior Angles With x

The 4-mark question: a trapezium has one pair of parallel sides. The two angles between them on the same side are co-interior.

In trapezium $A B C D$ , $A B$ is parallel to $D C$ . Angle $A$ is $(3 x + 10)^{\circ}$ and angle $D$ is $(2 x + 20)^{\circ}$ .

Worked example

1
$(3 x + 10) + (2 x + 20) = 180$ — AB is parallel to DC, so angles A and D are co-interior. Co-interior angles add up to 180°.
2
$5 x + 30 = 180$ — Collect like terms: 3x + 2x = 5x and 10 + 20 = 30.
3
$5 x = 150$ — Take 30 from both sides, so the x term is alone.
4
$x = 30$ — Divide both sides by 5. This is x, not an angle yet.
5
$∠ A = 3 (30) + 10 = 10 0^{\circ}$ — Put x = 30 back into the expression for angle A.
6
$∠ D = 2 (30) + 20 = 8 0^{\circ}$ — Put x = 30 into the expression for angle D.
7
$100 + 80 = 180$ — Check: the two angles add up to 180°, so they fit the co-interior rule.

Common mistake

Stopping at $x = 30$ when the question asks for an angle. Put $x$ back in.

Prove Two Lines Are Parallel

The 5-mark "show that" question works the rule backwards. If co-interior angles add to $18 0^{\circ}$ , the lines must be parallel.

On one line, two angles on a straight line are $(7 x + 5)^{\circ}$ and $(3 x + 25)^{\circ}$ . The angle co-interior with the second one is $(6 x + 20)^{\circ}$ .

Worked example

1
$(7 x + 5) + (3 x + 25) = 180$ — The two angles sit together on one straight line. Angles on a straight line add up to 180°.
2
$10 x + 30 = 180$ — Collect like terms: 7x + 3x = 10x and 5 + 25 = 30.
3
$10 x = 150$ — Take 30 from both sides, so the x term is alone.
4
$x = 15$ — Divide both sides by 10.
5
$3 x + 25 = 7 0^{\circ}$ — Put x = 15 into the second angle: 3 × 15 + 25 = 70. This is one angle of the pair.
6
$6 x + 20 = 11 0^{\circ}$ — Put x = 15 into its co-interior partner: 6 × 15 + 20 = 110.
7
$70 + 110 = 180$ — The co-interior angles add up to 180°, so the two lines are parallel.

Key idea

End with a sentence that says what you have shown. The last mark is for the conclusion.

Which Method? Read the Question

Find the transversal first. Then look at the shape the two angles make.

You see	You do
Same position at both crossings	corresponding: equal
Z shape between the lines	alternate: equal
C shape between the lines	co-interior: add to 180°
A triangle as well	Angles in a Triangle
Two lines meet on a parallel line	Z, straight line, then the triangle
Two pairs of parallel lines	jump one pair at a time
A bend between the lines	Multi-Step Angle Reasoning
Show the lines are parallel	find both angles, test the rule
Many steps, reasons for each	Multi-Step Angle Reasoning

Key idea

Stuck? Extend the lines on the diagram with a ruler. The F, Z or C often appears.

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