Multi-Step Angle Reasoning

Chain angle facts together, one step at a time, and write the exact reason for each stage, including exterior angles, reflex angles and bends between parallel lines.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 17 real past-paper questions below

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A Reason for Every Line

The 4 or 5-mark question says "Give a reason for each stage of your working". Find any angle you can, write its reason, then use it to find the next one.

Fact	Reason to write
On a line	angles on a straight line add up to 180°
Round a point	angles at a point add up to 360°
Crossing lines	vertically opposite angles are equal
Triangle	angles in a triangle add up to 180°
A side extended past a triangle	the exterior angle of a triangle equals the sum of the two opposite interior angles
All three sides equal	angles in an equilateral triangle are each 60°
Two equal sides	base angles of an isosceles triangle are equal
Four sides	angles in a quadrilateral add up to 360°
Parallel lines	alternate or corresponding angles are equal
Parallel lines, C shape	co-interior angles add up to 180°

Key idea

Mark each angle you find on the diagram as you go. The next step is often next to it.

Key idea

Often 2 of the marks are for the reasons. They only count when the working next to them is right, so find the angle carefully first.

Common mistake

Writing “Z angles”, “F angles” or just “opposite”. Use the full words: alternate, corresponding, vertically opposite. The reason mark needs them.

The Exterior Angle of a Triangle

Extend one side of a triangle. The angle outside is the exterior angle. It equals the two inside angles that are furthest from it.

Triangle $P QR$ has angle $P = 7 2^{\circ}$ and angle $Q = 5 3^{\circ}$ . Side $QR$ is extended to $S$ . Find angle $P R S$ .

Worked example

1
$∠ P = 7 2^{\circ}, ∠ Q = 5 3^{\circ}$ — Pick the two inside angles that do not touch R. The exterior angle at R is built from these two.
2
$∠ P R S = 72 + 53$ — The exterior angle of a triangle equals the sum of the two opposite interior angles. Write this reason in full.
3
$∠ P R S = 12 5^{\circ}$ — 72 + 53 = 125. It is less than 180°, as it must be, since it sits on the straight line QRS.

Key idea

It is a shortcut for two steps: $180 - 72 - 53 = 55$ , then $180 - 55 = 125$ .

Common mistake

Adding the angle next to the exterior angle. Use the two angles that do not touch it.

Crossing Lines in the Chain

Two straight lines crossing make an X. The angles opposite each other are equal, and this is often the first link in the chain.

Straight lines $A B$ and $C D$ cross at $P$ . Angle $A P C = 6 4^{\circ}$ . In triangle $P B D$ , angle $P D B = 4 7^{\circ}$ . $G$ is on $A B$ extended past $B$ . Find angle $D B G$ .

Worked example

1
$∠ B P D = 6 4^{\circ}$ — Vertically opposite angles are equal. APC and BPD sit opposite each other where AB and CD cross.
2
$∠ P B D = 180 - 64 - 47$ — Angles in a triangle add up to 180°. Triangle PBD now has two known angles.
3
$∠ P B D = 6 9^{\circ}$ — 180 − 64 = 116, then 116 − 47 = 69.
4
$∠ D B G = 180 - 69$ — Angles on a straight line add up to 180°. PB and BG lie on the same line AB.
5
$∠ D B G = 11 1^{\circ}$ — 180 − 69 = 111. Check with the exterior angle: 64 + 47 = 111 too.

Common mistake

Writing “opposite angles are equal”. The reason must say vertically opposite to get the mark.

A Reflex Angle in the Diagram

The 4-mark question can give a reflex angle. Take it from $36 0^{\circ}$ to get the angle inside the shape.

In triangle $A B C$ , the reflex angle at $B$ is $29 0^{\circ}$ and angle $A = 4 5^{\circ}$ . $A C D$ is a straight line. Find angle $B C D$ .

Worked example

1
$∠ A B C = 360 - 290$ — Angles at a point add up to 360°. The reflex angle and the angle inside the triangle go all the way round B.
2
$∠ A B C = 7 0^{\circ}$ — 360 − 290 = 70. It is less than 180°, so it can be an angle of the triangle.
3
$∠ B C D = 70 + 45$ — ACD is a straight line, so BCD is an exterior angle. It equals the two opposite interior angles, at B and A, added.
4
$∠ B C D = 11 5^{\circ}$ — 70 + 45 = 115.

Common mistake

Using 290° inside the triangle. A triangle's angles are each less than 180°.

When the Top Angle Is Given

Two equal sides, and the question gives the angle between them. The other two angles share what is left of 180° equally.

Formula

base angle = \frac{18 0 ^{\circ} - top angle}{2}

The top angle is the one between the two equal sides. The two base angles are at the ends of the third side, and they are equal.

Triangle $A B E$ has $A B = A E$ and angle $B A E = 3 6^{\circ}$ . It sits on side $B E$ of rectangle $B C D E$ . Find the reflex angle $x$ at $E$ , outside the shape, between $E A$ and $E D$ .

Worked example

1
$180 - 36 = 144$ — Angles in a triangle add up to 180°. Take away the top angle: the two base angles share the 144° left.
2
$∠ A E B = 144 \div 2 = 7 2^{\circ}$ — Base angles of an isosceles triangle are equal, so each one gets half.
3
$∠ B E D = 9 0^{\circ}$ — Every angle of a rectangle is 90°.
4
$x = 360 - 72 - 90$ — Angles at a point add up to 360°. The angles AEB, BED and x go all the way round E.
5
$x = 19 8^{\circ}$ — 72 + 90 = 162, and 360 − 162 = 198. It is more than 180°, which is right, because x is a reflex angle.

Common mistake

Forgetting to halve, so each base angle comes out as 144°. Two angles of 144° add to more than 180°, so that cannot be right.

A Chain of Four Facts

The 5-mark question: a straight line, an isosceles triangle and a quadrilateral all in one diagram.

$D E F$ is a straight line with angle $C E F = 11 8^{\circ}$ . $C D = C E$ . In quadrilateral $A B C D$ , $A = 8 8^{\circ}$ , $B = 10 4^{\circ}$ and $∠ A D C = 9 5^{\circ}$ . Find angle $B C E$ .

Worked example

1
$∠ C E D = 180 - 118$ — Angles on a straight line add up to 180°. CED and CEF sit side by side on the line DEF.
2
$∠ C E D = 6 2^{\circ}$ — 180 − 118 = 62.
3
$∠ C D E = 6 2^{\circ}$ — Base angles of an isosceles triangle are equal. CD = CE, so the angles at D and E match.
4
$∠ D C E = 180 - 62 - 62$ — Angles in a triangle add up to 180°. Triangle CDE now has two known angles.
5
$∠ D C E = 5 6^{\circ}$ — 62 + 62 = 124, and 180 − 124 = 56. This is one part of the angle at C.
6
$∠ B C D = 360 - 88 - 104 - 95$ — Angles in a quadrilateral add up to 360°. Use the three known angles of ABCD.
7
$∠ B C D = 7 3^{\circ}$ — 88 + 104 + 95 = 287, and 360 − 287 = 73. This is the other part of the angle at C.
8
$∠ B C E = 73 + 56 = 12 9^{\circ}$ — BCD and DCE sit side by side and make up BCE, so add the two parts.

Common mistake

Writing only the numbers. Without the reasons, a 5-mark answer can lose 2 or 3 marks.

A Bend Between Parallel Lines

The 4-mark question: a path bends at $E$ , between two parallel lines. Draw a new line through $E$ , parallel to both. Now there are two Z shapes.

$A B$ is parallel to $C D$ . Angle $B A E = 4 8^{\circ}$ and angle $A E C = 11 5^{\circ}$ . Find angle $E C D$ .

Worked example

1
$draw a line through E ∥ A B$ — This new line is parallel to both AB and CD. It splits the 115° at E into a top part and a bottom part.
2
$top part = 4 8^{\circ}$ — Alternate angles are equal. The top part and BAE make a Z shape with AB and the new line.
3
$bottom part = 115 - 48$ — The two parts together make the whole 115° at E.
4
$bottom part = 6 7^{\circ}$ — 115 − 48 = 67.
5
$∠ E C D = 6 7^{\circ}$ — Alternate angles are equal. The bottom part and ECD make a Z shape with the new line and CD.

Key idea

The bend angle is always the two outer angles added: $48 + 67 = 115$ .

Which Method? Read the Question

List what the question tells you: straight lines, equal sides, parallel lines. Each one is a fact you can use.

You see	You do
A side extended past a triangle	add the two opposite inside angles
A reflex angle	360 − it, to get the inside angle
Two lines crossing (an X)	vertically opposite angles are equal
Two equal sides	isosceles: two equal angles
Two equal sides, top angle given	(180 − top angle) ÷ 2 for each base angle
A bend between parallel lines	draw a parallel line through the bend
Only lines and a point	Angles at a Point and on a Line
One polygon, many sides	Interior and Exterior Angles of Polygons

Key idea

Write each reason in full, in words, next to its line.

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