Congruence and Similarity
Print two copies of the same photo and they match perfectly. Shrink one to fit a phone screen and it's smaller, but still clearly the same picture. Stretch it sideways and faces suddenly look wrong. Those three cases are what this page is about: shapes that are exactly the same (congruent), shapes that are the same but resized (similar), and shapes that are neither.
National curriculum KS3: use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles; identify and construct congruent triangles, and construct similar shapes by enlargement; apply triangle congruence and similarity to derive results about angles and sides1 · Congruent: same shape, same size
Think of two £1 coins, two pages from a photocopier, or two tiles from the same box. If you put one on top of the other, it covers it perfectly, with nothing sticking out. Mathematicians call shapes like that congruent (say "con-GROO-ent").
Here is the important part: to check, you're allowed to pick the shape up and move it. You may slide it, turn it or even flip it over. What you may not do is stretch it or shrink it. So a shape is still congruent to its mirror image, and to a copy that is lying at a different angle.
Try it. Drag the teal triangle to slide it. Use the buttons to turn it a quarter turn or flip it over. Can you make it land exactly on the dashed purple triangle?
Now switch to piece B. It looks almost the same, and from across a room you might say it matches. But no amount of sliding, turning or flipping makes it fit, because one of its sides is a different length. "Looks about the same" is not congruent. Congruent means exactly the same lengths and exactly the same angles.
What doesn't matter: where the shape is, which way round it faces, whether it's been flipped, and what colour it is.
What does matter: every length and every angle. If all of those match, the shapes are congruent.
2 · Matching parts: corresponding sides and angles
When one congruent shape sits on top of another, every side lands on a partner side and every corner lands on a partner corner. Those partners are called corresponding sides and corresponding angles. In congruent shapes, corresponding sides are equal and corresponding angles are equal.
Triangles have a standard way of being labelled. The corners (called vertices; one is a vertex) get capital letters, like A, B and C. The side joining A and B is called AB. The angle at corner A is written ∠A (you may also see ∠BAC, which means "the angle at A, between B and C").
The symbol ≅ means "is congruent to". When you write ABC ≅ PQR the order of the letters is a promise: A matches P, B matches Q and C matches R. So side BC (2nd and 3rd letters) matches side QR (2nd and 3rd letters). Pick a part of triangle ABC to see its partner.
Common mistake: matching the wrong sides. Triangle PQR has been turned and flipped, so "the bottom side" of one is not the bottom side of the other. Don't match by position on the page. Match by the letters (BC ↔ QR), or by size: the shortest side matches the shortest side, the longest matches the longest, and the angle opposite a side matches the angle opposite its partner.
3 · Four tests for congruent triangles
A triangle has six measurements: three sides and three angles. To prove two triangles are congruent, do you have to check all six? No. Imagine describing a triangle over the phone so that a friend draws exactly the same one. Some sets of just three facts are enough, because they lock the triangle into one shape and size. These are the four congruence conditions:
| Test | What you know | Example |
|---|---|---|
| SSS | All three Sides | 7 cm, 6 cm, 5 cm |
| SAS | Two sides and the angle between them (the included angle) | 7 cm, 50°, 5 cm |
| ASA / AAS | Two angles and one side, as long as it's a matching (corresponding) side | 50°, 7 cm, 60° |
| RHS | Right angle, Hypotenuse (the longest side, opposite the right angle) and one other Side | 90°, 8 cm, 5 cm |
Why is AAS allowed? Angles in a triangle add up to 180°, so if you know two angles you know the third. Two angles and any side is really the same as ASA, provided the sides you compare are corresponding ones.
Use the builder below. Pick a test and change the measurements. The dashed lines show how you'd construct it with a ruler, protractor and compasses. Each time, the given facts (in orange) leave exactly one place for corner C.
Whatever you choose, you only ever get one triangle (or none, if the numbers are impossible). That's why these tests work: two triangles that share the same SSS, SAS, ASA or RHS facts must be the same triangle, so they are congruent, and then all their other sides and angles match too.
4 · Why SSA and AAA don't work
Two other combinations of three facts look like they should work, but don't. The first is SSA: two sides and an angle that is not between them. Below, you know side AB, the angle at A, and side BC. Corner C has to be on the dashed line from A (because of the angle) and on the dashed circle around B (because BC has a fixed length). Watch what happens.
With AB = 8 cm, ∠A = 35° and BC = 5 cm, the circle crosses the line twice, so there are two completely different triangles with those same three facts. This is called the ambiguous case. A test for congruence has to work every time, and SSA sometimes gives two answers, so SSA can't prove two triangles are congruent. (The exception is when the known angle is 90°: then the circle can only land once, and that is exactly the RHS test.)
The second is AAA: all three angles. Angles fix the shape but say nothing about the size. Change the size below: the angles never change, but the sides do.
Common mistake: "same angles, so congruent". Two triangles with angles 40°, 60° and 80° could be the size of a stamp or the size of a football pitch. Same angles means the same shape, which is called similar (next section). It only means congruent if at least one pair of matching sides is also equal.
5 · Similar: same shape, different size
When you pinch-zoom a photo on a phone, it gets bigger but still looks right: faces aren't squashed and circles stay round. When you drag only one edge of a picture in a document, it gets stretched, and it looks wrong. The zoomed photo is similar to the original. The stretched one is not.
Two shapes are similar if one is an enlargement of the other: every length is multiplied by the same number, called the scale factor. When that happens, every angle stays exactly the same. (Enlargement, maps and how areas grow are covered in more depth on the Scale Factors and Drawings page.)
Below, the purple house is a copy of the teal one. With "keep in proportion" ticked, width and height grow by the same factor. Untick it and stretch just one direction. Watch the angle at the top of the roof and the round window.
So two shapes are similar when both of these are true:
- all corresponding angles are equal, and
- all corresponding sides are in the same ratio: dividing each side of the big shape by its partner on the small shape always gives the same answer, the scale factor.
Congruent shapes are a special case of similar shapes: the scale factor is 1.
6 · Scale factor and missing lengths
To find the scale factor, take a pair of corresponding sides where you know both lengths and divide:
scale factor = new length ÷ original length
A scale factor bigger than 1 makes the shape bigger; a scale factor between 0 and 1 makes it smaller. Slide k and look at the table. Triangle PQR is similar to triangle ABC, with A ↔ P, B ↔ Q, C ↔ R. Matching angles are marked with the same number of arcs.
Common mistake: adding instead of multiplying. Look at the last column of the table. The ÷ column is the same for every side; the − column is different for every side. If AB = 4 becomes PQ = 8, you can't say "it went up by 4, so BC = 3 becomes 7". It went times 2, so BC = 3 becomes 6. Adding the same amount to every side changes the shape.
The method
- Match the sides. Use the letters, or the angles, to decide which side corresponds to which. Redraw the triangles the same way round if that helps.
- Find the scale factor from a pair where you know both lengths: big ÷ small.
- Multiply by the scale factor to go from the small shape to the big one, or divide to go from big to small.
- Sense check: a side on the bigger shape must come out bigger than its partner.
Now practise. Find x, then check. If you go wrong, the feedback tries to spot which mistake you made.
7 · Similar triangles hiding inside triangles
Exam questions often hide two similar triangles inside one picture. The trick is a pair of parallel lines (lines that go in the same direction and never meet). Draw a line DE across triangle ABC, parallel to BC. That makes a small triangle ADE at the top.
Why are ADE and ABC similar? Check the angles:
- ∠A is shared: it is the same angle in both triangles.
- ∠ADE = ∠ABC, because they are corresponding angles on parallel lines (the "F-angle" rule).
- ∠AED = ∠ACB for the same reason.
All three angles match, so the triangles are similar. (AAA can't prove congruence, but it is enough to prove similarity.) Slide DE up and down, then tick "pull apart" to see the two triangles separately.
Common mistake: using DB instead of AB. The big triangle's side is the whole of AB, which is AD + DB. DB on its own isn't a side of either triangle, so comparing AD with DB gives the wrong scale factor. Always pull the triangles apart (or sketch them separately) before you match sides.
8 · Congruence, similarity and transformations
A transformation is a rule for moving or changing a shape. The shape you start with is the object and the result is the image. The moves you used in section 1 are transformations: sliding is a translation, turning is a rotation and flipping is a reflection. None of them change any length or angle, so the image is always congruent to the object.
An enlargement multiplies every length by the scale factor and keeps the angles, so the image is similar to the object (see Scale Factors and Drawings). A stretch in just one direction changes the angles, so the image is neither.
| Move | Lengths | Angles | Image |
|---|---|---|---|
| Translation, rotation, reflection | unchanged | unchanged | congruent (and similar, scale factor 1) |
| Enlargement, scale factor k | all × k | unchanged | similar |
| Stretch in one direction | some change, some don't | changed | neither |
Worked examples
Triangle ABC has AB = 7 cm, BC = 5 cm and ∠B = 40°. Triangle PQR has PQ = 7 cm, QR = 5 cm and ∠Q = 40°. Are they congruent? If AC = 4.6 cm, what is PR?
Triangle XYZ has XY = 8 cm, YZ = 5 cm and ∠X = 35°. Triangle LMN has LM = 8 cm, MN = 5 cm and ∠L = 35°. Must they be congruent?
Triangles ABC and DEF are similar, with A ↔ D, B ↔ E, C ↔ F. AB = 4 cm, BC = 6 cm, AC = 5 cm and DE = 10 cm. Find EF and DF.
A photo measures 24 cm by 18 cm. A similar, smaller print has its long side 8 cm. How long is its short side?
In triangle ABC, D is on AB and E is on AC, with DE parallel to BC. AD = 3 cm, DB = 6 cm, DE = 4 cm and AE = 2 cm. Find BC and EC.
Quick check
What you've learned
- Congruent shapes are the same shape and the same size. One fits exactly on the other after sliding, turning or flipping. Mirror images count.
- Corresponding sides and angles are the matching parts. In ABC ≅ PQR the letter order tells you: A ↔ P, B ↔ Q, C ↔ R, so BC ↔ QR.
- Two triangles are congruent if they share SSS, SAS (included angle), ASA/AAS (matching side) or RHS. Each of these fixes exactly one triangle.
- SSA doesn't prove congruence because it can give two different triangles (the ambiguous case). AAA doesn't either: it fixes the shape but not the size.
- Similar shapes are the same shape, different size: one is an enlargement of the other. Corresponding angles are equal and corresponding sides are all in the same ratio.
- Scale factor = new length ÷ original length, from a pair of corresponding sides. Multiply to go small → big, divide to go big → small. Never add.
- A line parallel to one side of a triangle cuts off a smaller similar triangle. Use whole sides (AB = AD + DB) when finding the scale factor.
- Translations, rotations and reflections give congruent images; enlargements give similar images.