1MA1 · G1, G3, G6Geometry & MeasuresFoundation & HigherGrades 1–5

Angles & Parallel Lines

Angle problems are logic puzzles: a few facts, applied in the right order, unlock every missing angle in a diagram. Edexcel does not just want the numbers — "give a reason for each stage of your working" means stating the angle fact used, in the correct words. This lesson covers the facts and the language.

45 min Angles & Geometric Reasoning

The big picture

Everything rests on a small set of facts: angles on a straight line sum to $180^{\circ}$ , angles around a point to $360^{\circ}$ , vertically opposite angles are equal, and angles in a triangle sum to $180^{\circ}$ . Parallel lines add three more: alternate angles are equal, corresponding angles are equal, and co-interior angles sum to $180^{\circ}$ .

Reasons earn marks independently of the numbers. "Alternate angles are equal" earns the reasoning mark; "Z angles" does not. Learning the precise phrases is as important as spotting the angles.

What you'll learn

Use angle facts on straight lines, around points and in triangles
Identify alternate, corresponding and co-interior angles on parallel lines
Solve multi-step angle problems, including with algebra
Give geometric reasons using correct mathematical language

Basic angle facts

Angles on a straight line sum to $180^{\circ}$ . Angles around a point sum to $360^{\circ}$ .

Vertically opposite angles — formed where two straight lines cross — are equal.

Angles in a triangle sum to $180^{\circ}$ . An isosceles triangle has two equal sides and two equal base angles. Each angle in an equilateral triangle is $60^{\circ}$ .

Angles in a quadrilateral sum to $360^{\circ}$ .

Example 1In isosceles triangle

ABC

AB = AC

and angle

BAC = 38^{\circ}

. Find angle

ABC

1Base angles are equal: angle

ABC

= angle

ACB

2 \times ABC = 180^{\circ} - 38^{\circ} = 142^{\circ}

AnswerAngle

ABC = 71^{\circ}

(angles in a triangle sum to

180^{\circ}

; base angles of an isosceles triangle are equal)

Your turnSame method, new numbers — try it before moving on.

An isosceles triangle has two equal base angles of $64^\circ$ . Find the third angle, in degrees.

Your answer

Show worked solution

Solution

1Angles in a triangle sum to $180^\circ$ .
2 $180 - 64 - 64 = 52^\circ$ .

Answer: $52^\circ$

Parallel lines

When a line (a transversal) crosses two parallel lines, it creates pairs of related angles.

$\textbf{Alternate angles}$ are equal — they sit on opposite sides of the transversal, between the parallel lines (a Z shape).

$\textbf{Corresponding angles}$ are equal — they are in matching positions at each crossing (an F shape).

$\textbf{Co-interior angles}$ sum to $180^{\circ}$ — they are on the same side of the transversal, between the parallel lines (a C or U shape).

Example 1Two co-interior angles between parallel lines are

(3x + 10)^{\circ}

and

(2x - 5)^{\circ}

. Find both angles.

1Co-interior angles sum to

180^{\circ}

5x + 5 = 180

x = 35

Answer

115^{\circ}

and

65^{\circ}

Your turnSame method, new numbers — try it before moving on.

Two corresponding angles on parallel lines are $(4x - 20)^\circ$ and $(2x + 30)^\circ$ . Find $x$ .

Your answer

Show hint

Corresponding angles are equal.

Show worked solution

Solution

1 $4x - 20 = 2x + 30$ .
2 $2x = 50$ , so $x = 25$ (each angle is $80^\circ$ ).

Answer: $x = 25$

Tip — Use the shapes to spot the angles, but write the names in answers: alternate, corresponding, co-interior.

Multi-step problems

Label the angles you find on the diagram as you go; each one may unlock the next.

Write one fact per line, with the angle and its reason. If the question says "give reasons", every step needs one.

Example 1

PQ

is parallel to

RS

. A transversal meets

PQ

X

and

RS

Y

. Angle

QXY = 72^{\circ}

. Point

Z

RS

is such that triangle

XYZ

is isosceles with

XY = XZ

. Find angle

YXZ

1Angle

XYZ = 72^{\circ}

: alternate angles are equal.

2Angle

XZY = 72^{\circ}

: base angles of an isosceles triangle are equal.

3Angle

YXZ = 180^{\circ} - 144^{\circ}

: angles in a triangle sum to

180^{\circ}

AnswerAngle

YXZ = 36^{\circ}

Why this works

A diagram labelled "not drawn accurately" means you cannot measure. Every angle must come from a fact — which is exactly why reasons are worth marks.

Proofs with angles

Angle facts can prove other facts. For example, the angles in a triangle sum to $180^{\circ}$ can be proved with a line through one vertex parallel to the opposite side: the two alternate angles and the third angle make a straight line.

The exterior angle of a triangle equals the sum of the two opposite interior angles — a useful shortcut that follows from the angle sum.

\text{exterior angle} = \text{sum of opposite interior angles}

Exterior angle of a triangle.

Think like an examiner

State a reason for each step, using full phrases: "alternate angles are equal".
Do not measure angles on diagrams marked "not drawn accurately".
Mark found angles on the diagram to keep track.

Angle facts

180^{\circ}

Straight line; triangle.

360^{\circ}

Around a point; quadrilateral.

a + b = 180^{\circ}

Co-interior angles.

Watch out for these

Writing "Z angles" as a reason.

Write "alternate angles are equal".

Assuming co-interior angles are equal.

They sum to

180^{\circ}

Using parallel line facts when lines are not marked parallel.

Only use them when the lines are stated or marked parallel.

Assuming which angles are equal in an isosceles triangle without checking the equal sides.

The equal angles are opposite the equal sides.

Stretch yourself

Prove that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Hint — Call the interior angles $a$ , $b$ and $c$ , with the exterior angle next to $c$ . Use two different facts involving $c$ .

Reveal solution

SolutionLet the interior angles be

a

b

and

c

, and let the exterior angle at the vertex with angle

c

e

. Angles in a triangle sum to

180^{\circ}

, so

a + b + c = 180^{\circ}

. Angles on a straight line sum to

180^{\circ}

, so

c + e = 180^{\circ}

. Therefore

a + b + c = c + e

, and subtracting

c

gives

e = a + b

\blacksquare

Questions students ask

How do I tell alternate from corresponding angles?

Alternate angles are on opposite sides of the transversal and between the parallel lines. Corresponding angles are in the same relative position at each intersection.

Do I always need to give reasons?

Only when asked, but it is good practice — and many questions do ask.

What does "not drawn accurately" mean?

You must calculate angles using facts, not measure them with a protractor.

Key takeaways

Straight line $180^{\circ}$ ; around a point $360^{\circ}$ ; vertically opposite angles equal.
Triangle $180^{\circ}$ ; quadrilateral $360^{\circ}$ .
Parallel lines: alternate equal, corresponding equal, co-interior sum to $180^{\circ}$ .
Give a reason in full words for each step.
Exterior angle of a triangle equals the sum of the opposite interior angles.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

Three angles on a straight line are $45^\circ$ , $x$ and $2x$ . Work out the value of $x$ .[2]
Show mark scheme
- M1 $45 + 3x = 180$
- A1 $x = 45$
Answer: $x = 45$

Question 2Grade 42 marks

$AB$ is parallel to $CD$ . A straight line crosses $AB$ at $B$ and $CD$ at $C$ . Angle $ABC = 58^\circ$ , and angle $BCD$ is on the other side of $BC$ .

Work out angle $BCD$ . Give a reason for your answer.[2]
Show mark scheme
- B1 $58^\circ$
- B1Alternate angles are equal
Answer: $58^\circ$

Question 3Grade 53 marks

Prove that the angles in any triangle add up to $180^\circ$ .[3]
Show mark scheme
- B1Draws a line through one vertex parallel to the opposite side
- B1Identifies two pairs of alternate angles
- C1Uses angles on a straight line summing to $180^\circ$ to complete the proof

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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