MathsGCSE maths: geometry, Pythagoras and trigonometry30 min★★★ difficulty

Angle reasoning, polygons and similarity

Geometry marks are given for reasons as much as answers. Each step should name the fact that justifies it.

Lesson overview

What you'll learn in this lesson

Reason geometrically and calculate lengths, angles, areas and volumes.

Key learning points

• Angle facts to quote by name
• Polygons
• Congruence and similarity

This lesson at a glance

30 minutes
16 parts to scroll through
4 quick checks
Marked quiz at the end
Gentle pace: short sittings with pauses

Words to know

reasongeometricallycalculatelengthsangles

Scroll down — the lesson carries on below

Part 1 of 16 · Discover6%

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Part 1 of 16

Watch first

Angle reasoning, polygons and similarity

An English teaching video on this topic from Cognito. Watch it, then scroll down for the lesson.

In a nutshell

Reason geometrically and calculate lengths, angles, areas and volumes.

Learning cycle

Part 2 of 16

Learning cycle 1 of 2

Part 1 · Angle facts to quote by name

A short piece of teaching, then a check to make sure it has landed.

Explore the idea

Part 3 of 16

Learn

Angle facts to quote by name

Angles on a straight line total 180°, around a point 360°, and in a triangle 180°. With parallel lines, alternate angles are equal, corresponding angles are equal and co-interior angles sum to 180°. Write the name, not just the number: 'alternate angles are equal' earns the reasoning mark.

Explore the idea

Part 4 of 16

Learn

Polygons

The exterior angles of any polygon sum to 360°, so a regular polygon with n sides has exterior angle 360 ÷ n. The interior angle sum is (n − 2) × 180°. Working with exterior angles is usually faster: a regular decagon has exterior angle 36° and therefore interior angle 144°.

Quick check

Part 5 of 16

Quick check

Co-interior angles between parallel lines

Quick check

Part 6 of 16

Quick check

A regular octagon has exterior angle

Reset break

Part 7 of 16

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now

Learning cycle

Part 8 of 16

Learning cycle 2 of 2

Part 2 · Congruence and similarity

A short piece of teaching, then a check to make sure it has landed.

Explore the idea

Part 9 of 16

Learn

Congruence and similarity

Congruent triangles are proved by SSS, SAS, ASA or RHS, and you must state which. Similar shapes have equal angles and sides in a fixed ratio. For a length scale factor k, areas scale by k² and volumes by k³ — the fact behind many multi-mark questions.

Quick check

Part 10 of 16

Quick check

If lengths scale by 3, volumes scale by

Quick check

Part 11 of 16

Quick check

RHS congruence applies to

Explore the idea

Part 12 of 16

Worked example

Worked answer: a regular polygon has an interior angle of 156°. How many sides? (3 marks)

The exterior angle is 180 − 156 = 24°, because interior and exterior angles lie on a straight line (1). The exterior angles of any polygon sum to 360° (1), so n = 360 ÷ 24 = 15 sides (1). Working from the interior angle sum formula reaches the same answer but takes longer and invites arithmetic slips.

Challenge round

Part 13 of 16

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

Challenge round

Part 14 of 16

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

____ on a straight line total 180°, around a point 360°, and in a triangle 180°.

Mastery quiz

Part 15 of 16

Marked quiz

End of lesson quiz: Angle reasoning, polygons and similarity

4 questions, marked with the reasoning shown. No timer.

1. Co-interior angles between parallel lines
2. A regular octagon has exterior angle
3. If lengths scale by 3, volumes scale by
4. RHS congruence applies to

Lesson round-up

Part 16 of 16

Lesson round-up

Ready when you are

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Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

Work out the total cost of the trip if 50 pupils go.
Show why the price per pupil falls when more pupils go, using numbers.
At £20 each, how many pupils must go before the trip makes money? Show your working.

Working out
Working out
Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
Writing
Copy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working out
Write your own real-life problem that uses today's method, then solve it.
1 problem + solution

Write your answer, notes or working here

Saved as you type.

Photos of paper work, diagrams or working outHow many minutes did the written work take?

Part 1 of 16 · Watch & discover

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