Angle Properties — AQA GCSE Mathematics

SPEC 4.01 • G1, G3 — ANGLE PROPERTIES

Angles at a Point, a Line, Vertically Opposite & Triangles

The angle facts every GCSE student needs — and why stating the reason is worth as many marks as the angle itself.

Why Reasons Matter

Geometry questions award marks for two things:

Mark 1 — Value

The correct numerical answer for the angle you are asked to find.

Mark 2 — Reason

The named geometric fact you used to get it.

AQA Rule

A correct angle without a stated reason loses the communication mark — even when the number is right. Vague phrases like "angles add to 180°" are not accepted because they don't specify where (on a line? in a triangle? in a quadrilateral?).

Angle-Labelling Convention

Angles can be written in two ways:

Three letters — e.g. angle ABC. The middle letter is the vertex; the outer two give the arms. Angle ABC means the angle at vertex B formed by the arms BA and BC.

Single letter — e.g. angle x or angle B. Only used when it is unambiguous which angle is meant.

"Not Drawn Accurately"

Every AQA geometry diagram is printed with the instruction "Not drawn accurately". Never try to measure an angle from the page — use the angle facts and stated values. The diagram shows the configuration; the question gives the numbers.

Types of Angles

Before applying angle facts, you must be able to classify any angle you see.

Quick Classification

Always read the size of the angle first. If a question asks for the "reflex angle at P," the answer is bigger than 180° — never quote the acute one by mistake.

Angles at a Point = 360°

The angles around any single point always sum to 360° — one full turn.

∑ angles around a point = 360°

Mark-Scheme Wording

The accepted reason is "angles around a point = 360°" (or "angles at a point sum to 360°"). Saying "all angles add up to 360" without where loses the reason mark.

Angles on a Straight Line = 180°

When rays meet on one side of a straight line, the angles on that side sum to 180° — a half turn.

∑ angles on a straight line = 180°

Accepted Reason

"Angles on a straight line = 180°" or "angles on a straight line sum to 180°". Both score the reason mark in AQA papers.

Vertically Opposite Angles

When two straight lines cross, the pairs of angles opposite each other at the crossing point are equal.

Why it works

Each adjacent pair sits on a straight line, so each pair sums to 180°. That forces the opposite pairs to match.

Reason to write

"Vertically opposite angles are equal." That exact phrase earns the reason mark.

Worked Example — Angles at a Point01

Three rays meet at a point P. Two of the angles are 140° and 85°. Work out the size of the third angle. Give a reason for your answer. [2 marks]

M

Method: All angles at point P sum to 360°. Subtract the two known angles.

C

Calculate: x = 360 − 140 − 85 = 135°

E

Evaluate / Answer: x = 135° (angles around a point = 360°).

Mark split: B1 for 135, B1 for the named reason. Drop the reason and you drop the second mark.

Triangle Types You Must Know

Before applying the triangle angle-sum rule, recognise which type of triangle you are dealing with. AQA spec G4 lists these by name.

NameKey propertyAngle consequence

EquilateralAll three sides equalAll three angles = 60°

IsoscelesExactly two sides equalTwo angles equal (opposite the equal sides)

ScaleneAll sides differentAll three angles different

Right-angledOne angle = 90°The other two sum to 90°

Don't Assume

A diagram can look isosceles without being labelled as one. Only use the isosceles rule if the question either states two sides (or two angles) are equal, or marks them equal on the diagram (tick marks on sides).

Angles in a Triangle = 180°

The three interior angles of any triangle — whatever its shape — sum to 180°.

a + b + c = 180°

Accepted Reason

Write "angle sum of a triangle = 180°" or "angles in a triangle sum to 180°". Either version scores. The crucial word is triangle — without it, the reason is ambiguous and earns nothing.

Why This Is Foundational

Nearly every multi-step geometry question eventually uses this fact. Internalise it: the three interior angles of any triangle — acute, right, obtuse, isosceles, equilateral — always add to 180°.

Isosceles Triangles

An isosceles triangle has two equal sides. The angles opposite those equal sides — called the base angles — are equal.

The Rule

If AB = AC, then angle B = angle C. You identify the equal angles from the equal sides — not the other way round.

Reason Wording

Accepted phrase: "base angles of an isosceles triangle are equal". Variations like "it's isosceles so two angles are the same" are too loose for AQA and lose the reason mark.

MCE Worked Example — Isosceles Triangle02

In triangle ABC, AB = AC and angle A = 40°. Work out the size of angle B. Give reasons for your answer. [3 marks]

M

Method: Two equal sides ⇒ isosceles, so base angles B and C are equal. Let each equal x. Use angle sum of triangle = 180°.

C

Calculate:

40 + x + x = 180 → 2x = 140 → x = 70

E

Evaluate / Answer: Angle B = 70° (base angles of an isosceles triangle are equal; angle sum of a triangle = 180°).

Mark split: M1 for setting up 40 + 2x = 180 (or equivalent), A1 for x = 70, B1 for both named reasons. Missing the isosceles reason (or the triangle-sum reason) forfeits the B1.

Exterior Angle of a Triangle

Extend one side of a triangle past a vertex. The angle formed outside the triangle at that vertex is the exterior angle.

exterior angle = sum of the two opposite interior angles

Why it's true

Interior b + exterior = 180° (straight line).

And a + b + c = 180° (triangle).

Subtracting: exterior = a + c.

When to use it

Whenever you can see a triangle with one side extended. It turns a two-step calculation into a one-step shortcut.

Worked Example — Exterior Angle03

In triangle PQR, side PQ is extended to a point S. Angle SQR = 130° and angle QPR = 45°. Find angle QRP. Give a reason for your answer. [2 marks]

M

Method: The 130° is the exterior angle at Q. Exterior angle = sum of the two opposite interior angles.

C

Calculate: 130 = 45 + x → x = 130 − 45 = 85

E

Evaluate / Answer: Angle QRP = 85° (exterior angle of a triangle equals the sum of the two opposite interior angles).

Alternative: Interior angle at Q = 180 − 130 = 50° (angles on a straight line). Then x = 180 − 45 − 50 = 85° (angles in a triangle sum to 180°). Both methods score full marks — the exterior shortcut is simply faster.

Combining the Facts

Most exam questions chain two or three of these facts together. The skill is choosing the right fact at each step and stating its reason every time.

Strategy for multi-step angle questions

1

Find any angle you can from the information given. Label it on the diagram with its value.

2

Write the reason for that angle immediately , in brackets.

3

Use that new angle to unlock the next one. Again: value, reason.

4

Keep going until you hit the angle the question asked for.

Pro Tip — Annotate the Diagram

Writing each value on the diagram as you find it is one of the fastest ways to score method marks, even if you lose track of the algebra. AQA mark schemes explicitly allow working to be shown on the diagram.

The 6 Facts to Pick From

Angles at a point = 360°

Angles on a straight line = 180°

Vertically opposite angles are equal

Angle sum of a triangle = 180°

Base angles of an isosceles triangle are equal

Exterior angle of a triangle = sum of two opposite interior angles

04Worked Example — Two-Step Problem

In the diagram, ABC is a straight line. Triangle BCD has angle BDC = 110° and BD = CD. Work out the size of angle x (angle DBA). Give reasons for your answer. [3 marks]

(Diagram note: the apex angle at D is 110°; BD = CD, shown by the matching tick marks.)

M

Method: Two steps. First find angle DBC using the isosceles triangle. Then use angles on straight line ABC.

C1

Step 1 — angle DBC:

BD = CD so the triangle is isosceles; angles DBC and DCB are equal.

Angle DBC + angle DCB + 110 = 180 (angle sum of triangle)

2 × angle DBC = 70 ⇒ angle DBC = 35°

C2

Step 2 — angle x:

ABC is a straight line, so angle DBA + angle DBC = 180°

x = 180 − 35 = 145°

E

Evaluate / Answer: x = 145°

Reasons: base angles of an isosceles triangle are equal; angle sum of a triangle = 180°; angles on a straight line = 180°.

Mark split: M1 for using the triangle and isosceles facts to reach 35°, A1 for 145°, B1 for all three reasons cited. Correct 145° without the reason chain drops a mark.

Mark Scheme Language

Lock these phrases into memory. They score the reason mark every time.

FactReason to writeVague version that LOSES marks

Angles at a pointangles around a point = 360°"angles add up" (no location)

Angles on a lineangles on a straight line = 180°"angles make 180"

Vertically oppositevertically opposite angles are equal"X angles are equal"

Triangle angle sumangle sum of a triangle = 180° (or "angles in a triangle sum to 180°")"angles add to 180"

Isosceles base anglesbase angles of an isosceles triangle are equal"it's isosceles"

Exterior angleexterior angle of a triangle = sum of opposite interior angles"exterior angles add up"

How to Format Your Answer

Put the reason in brackets after the value, like this:

"Angle ABC = 70° (base angles of an isosceles triangle are equal)."

This is the exact formatting AQA mark schemes display in worked solutions.

Examiner Alerts

Alert 1 — Vague Reasons

Writing "angles add to 180°" is ambiguous — on a line? In a triangle? Inside a quadrilateral? AQA treats this as a non-reason and awards zero for the reason mark. Always specify where.

Alert 2 — Assuming Isosceles from a Diagram

A triangle that looks symmetric on the page might not be isosceles. Only apply the base-angles rule when the question states the equality (in words or with tick marks on the sides). Diagrams on AQA papers are labelled "Not drawn accurately" for exactly this reason.

Alert 3 — Giving the Reflex Instead of the Acute

When a question asks for "the reflex angle", the answer must be greater than 180°. Students often quote the acute angle (< 90°) by mistake and lose the mark. Read the question carefully.

Alert 4 — Value With No Reason on Multi-step Problems

AQA examiner reports note that students who reach the correct final angle with no reasoning chain regularly score only one mark out of four. Every intermediate angle you use needs a reason — not just the final answer.

Summary & Checkpoint

The six angle facts — memorise these reasons verbatim

Angles around a point = 360°

Angles on a straight line = 180°

Vertically opposite angles are equal

Angle sum of a triangle = 180°

Base angles of an isosceles triangle are equal

Exterior angle of a triangle = sum of two opposite interior angles

Exam technique

Every angle you calculate needs a named reason in brackets.

Annotate the diagram as you go — method marks can be earned on the figure.

For multi-step problems, work in short numbered steps, each with its reason.

Never assume isosceles or equilateral without explicit evidence on the diagram.

You should now be able to

Classify any angle as acute, right, obtuse, or reflex.

Calculate unknown angles at a point, on a line, or formed by intersecting lines.

Find missing angles in any triangle, including isosceles cases.

Apply the exterior-angle shortcut when a side is extended.

Combine two or three facts in a multi-step question — with a full reason at every step.

Next Lesson (4.02)

We extend these facts to parallel lines — alternate, corresponding, and co-interior angles — and learn the three AQA-specific reason phrases that replace "Z angles," "F angles," and "C angles."