Congruence, similarity and enlargement
Congruent shapes are identical; similar shapes are the same shape at a different size. Distinguishing the two, and proving which you have, is the heart of geometric reasoning.
Part of your national curriculum
- Geometry and measures: Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
Lesson overview
What you'll learn in this lesson
Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
Key learning points
- • Conditions for congruence
- • Enlargement from a centre
- • What enlargement preserves
- • Using similarity to find lengths
This lesson at a glance
- 30 minutes
- 21 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 21
Visual introductionPicture this
Congruence, similarity and enlargement
Congruent shapes are identical; similar shapes are the same shape at a different size. Distinguishing the two, and proving which you have, is the heart of geometric reasoning.
In a nutshell
Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
Learning cycle
Part 2 of 21
Learning cycle 1 of 2
Part 1 · Conditions for congruence
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 21
Learn
Conditions for congruence
Two triangles are congruent if they match on SSS, SAS, ASA or RHS. Note that SSA is not a condition: two sides and a non-included angle can produce two different triangles. Naming the condition you are using is required in a congruence proof.
Reset break
Part 4 of 21
Pause
That's sitting 1 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.
Explore the idea
Part 5 of 21
Learn
Enlargement from a centre
An enlargement needs a scale factor and a centre. From centre (0, 0) with factor 2, the point (3, 1) maps to (6, 2): each coordinate distance from the centre doubles. A fractional scale factor such as 1/2 produces a smaller image, and a negative factor puts the image on the opposite side of the centre.
Quick check
Part 6 of 21
Quick check
Part 7 of 21
Reset break
Part 8 of 21
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 21
Learning cycle 2 of 2
Part 2 · What enlargement preserves
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 10 of 21
Learn
What enlargement preserves
Enlargement preserves angles and the ratios of sides, so the image is similar to the object. Lengths change by k, areas by k² and volumes by k³. Describing a transformation fully means stating that it is an enlargement, giving the scale factor and giving the centre.
Explore the idea
Part 11 of 21
Learn
Using similarity to find lengths
In similar triangles, match corresponding vertices before writing any ratio. Draw both triangles the same way up, label matching sides, then form the scale factor from the pair where both lengths are known.
Reset break
Part 12 of 21
Pause
That's sitting 3 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.
Quick check
Part 13 of 21
Quick check
Part 14 of 21
Explore the idea
Part 15 of 21
Worked example
Worked answer: describe fully the transformation mapping (3, 1) to (6, 2) and (4, 3) to (8, 6) (3 marks)
Each coordinate has doubled, so the scale factor is 2 (1). Rays through each object point and its image meet at the origin, so the centre of enlargement is (0, 0) (1). The full description is: an enlargement, scale factor 2, centre (0, 0) (1). Naming only 'enlargement' without both the factor and the centre would score one mark at most.
Reset break
Part 16 of 21
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 21
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 18 of 21
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
Two triangles are ____ if they match on SSS, SAS, ASA or RHS.
Note that SSA is not a condition: two sides and a non-included angle can produce two different ____.
Enlargement preserves angles and the ratios of sides, so the image is ____ to the object.
Challenge round
Part 19 of 21
Game · Recall cards
Which condition proves two triangles are congruent?
Card 1 of 4
Mastery quiz
Part 20 of 21
Marked quiz
End of lesson quiz: Congruence, similarity and enlargement
4 questions, marked with the reasoning shown. No timer.
1. Which condition proves two triangles are congruent?
2. A triangle with sides 3, 4, 5 cm is enlarged by scale factor 2. What are the new sides?
3. What stays the same between similar shapes?
4. Which is NOT a congruence condition?
Lesson round-up
Part 21 of 21
Lesson round-up
Ready when you are
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Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids