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
- 20 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 20
Visual introductionPicture this
Angle reasoning, polygons and similarity
Geometry marks are given for reasons as much as answers. Each step should name the fact that justifies it.
In a nutshell
Reason geometrically and calculate lengths, angles, areas and volumes.
Learning cycle
Part 2 of 20
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 20
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.
Reset break
Part 4 of 20
Pause
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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.
Explore the idea
Part 5 of 20
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 6 of 20
Quick check
Part 7 of 20
Reset break
Part 8 of 20
Pause
That's sitting 2 of 5 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.
Learning cycle
Part 9 of 20
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 10 of 20
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 11 of 20
Reset break
Part 12 of 20
Pause
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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.
Quick check
Part 13 of 20
Explore the idea
Part 14 of 20
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 15 of 20
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
Reset break
Part 16 of 20
Pause
That's sitting 4 of 5 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.
Challenge round
Part 17 of 20
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°.
Challenge round
Part 18 of 20
Game · Recall cards
Co-interior angles between parallel lines
Card 1 of 4
Mastery quiz
Part 19 of 20
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 20 of 20
Lesson round-up
Ready when you are
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