Congruence and similarity is a core Sec 2 topic in both the G2 and G3 streams. Congruent figures have exactly the same shape and size. Similar figures have the same shape but may differ in size, because every length is multiplied by the same scale factor k.
The topic is the foundation for the formal congruence tests (SSS, SAS, ASA, RHS) and for the area and volume ratios of similar figures in Sec 3 and Sec 4. Most marks are lost in the set-up rather than the working: a wrong letter match or an upside-down ratio makes every line after it worthless. Enlargement and reduction is Sec 2 for G3 but Sec 3/4 for G2. Scale drawings are Sec 3/4 in both streams.
What these notes cover
- •Congruent figures and the ≅ symbol
- •Matching vertices from the letter order
- •Similar figures and the two similarity conditions
- •Why angles alone are enough for triangles but not quadrilaterals
- •Using the ratio chain to find an unknown side
- •Deciding whether two figures are similar
- •Overlapping triangles: the parallel-line case
- •The butterfly case
- •Shadow and lamp-post problems
- •Enlargement and reduction by a scale factor
- •Scale drawings and map scales (preview)
- •Common mistakes and exam technique
Key definitions
- Congruent figures
- Figures with exactly the same shape and exactly the same size. One can be placed on the other after a translation, rotation or reflection. All corresponding sides are equal and all corresponding angles are equal.
- Similar figures
- Figures with the same shape but not necessarily the same size, where one is an enlargement or reduction of the other. All corresponding angles are equal and all corresponding sides are in the same ratio k.
- Congruence or similarity statement
- A statement such as △ABC ≅ △PQR (congruent) or △ABC ∼ △XYZ (similar). The order of the letters fixes the matching: 1st letter with 1st, 2nd with 2nd and 3rd with 3rd.
- Enlargement
- A new figure (the image) in which every length of the original figure (the object) is multiplied by the same positive scale factor k. All angles stay the same and the image is similar to the object. k > 1 enlarges the figure, k = 1 gives a congruent image and 0 < k < 1 reduces it.
- Scale 1 : n
- On a scale drawing or map, 1 cm on the drawing represents n cm in real life. Both sides are in the same unit, so no units are written.
Key formulas
Congruent triangles
△ABC ≅ △PQR ⇒ AB = PQ, BC = QR, AC = PR and ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R
Read the letters in order: A ↔ P, B ↔ Q, C ↔ R. A side named by two letters matches the side formed by their partners.
Ratio chain for similar triangles
For △ABC ∼ △XYZ. Keep the same triangle on top in every fraction. k is the scale factor.
Parallel-line case (DE ∥ BC)
D is on AB, E is on AC and DE ∥ BC, so △ADE ∼ △ABC (AA). Use whole sides only, never the piece DB.
Butterfly case (AB ∥ DE)
AE and BD cross at C. △ACB ∼ △ECD by vertically opposite angles at C and alternate angles from AB ∥ DE.
Enlargement / reduction
image length = k × object length
Applies to every length: sides, heights and diagonals. Angles are never scaled.
Finding the scale factor
Use any pair of corresponding lengths, with the image on top. If k < 1, the figure has been reduced.
Worked example
In △ABC, D lies on AB and E lies on AC such that DE ∥ BC. It is given that AD = 4 cm, DB = 6 cm and DE = 3 cm. Find the length of BC.
- 1∠ADE = ∠ABC (corr. ∠s, DE ∥ BC) and ∠DAE = ∠BAC (common), so △ADE ∼ △ABC (AA).
- 2Find the whole side that matches AD: AB = AD + DB = 4 + 6 = 10 cm.
- 3Write the ratio of corresponding sides: DE ÷ BC = AD ÷ AB, so 3 ÷ BC = 4 ÷ 10.
- 4Cross-multiply: 4 × BC = 3 × 10 = 30, so BC = 30 ÷ 4 = 7.5 cm.
- 5Check: the scale factor from small to big is 10 ÷ 4 = 2.5, and 3 × 2.5 = 7.5 ✓.
Watch out: Using DB instead of AB gives 3 ÷ BC = 4 ÷ 6, so BC = 4.5 cm. This is too small, because DB is not a side of either triangle.
Common mistakes
Reading the match from the picture, not the letters
Wrong: In △PQR ∼ △XZY, deciding that Q matches Y because both sit at the bottom-right of the diagram.
Right: Write the pairs from the letter order first: P ↔ X, Q ↔ Z, R ↔ Y. This gives PQ ↔ XZ, QR ↔ ZY and PR ↔ XY.
Inverting one of the ratios
Wrong: AB ÷ XY = YZ ÷ BC, with one fraction big-over-small and the other small-over-big.
Right: AB ÷ XY = BC ÷ YZ = AC ÷ XZ. Every numerator comes from the same triangle.
Assuming equal angles is enough for quadrilaterals
Wrong: A square and a 3 × 5 rectangle both have four 90° angles, so they are similar.
Right: Polygons with four or more sides need both conditions. Here the side ratios are not equal, so the figures are not similar.
Questions students ask
What is the difference between congruent and similar figures?
Congruent figures have the same shape and the same size, so all corresponding sides and angles are equal. Similar figures have the same shape but may differ in size. Their corresponding angles are equal and their corresponding sides are in the same ratio k. Congruent figures are the special case of similar figures with k = 1.
How do you know which sides correspond in similar triangles?
Use the order of the letters in the statement, not the positions in the diagram. In △ABC ∼ △XYZ, A ↔ X, B ↔ Y and C ↔ Z, so AB matches XY, BC matches YZ and AC matches XZ. If no statement is given, pair the vertices with equal angles first, then write the statement in that order.
Are two shapes with equal angles always similar?
Only for triangles. Two pairs of equal angles is enough (AA), because the third pair follows from the angle sum of 180°. For quadrilaterals and other polygons, you must also check that all side ratios are equal. For example, a square and a non-square rectangle have equal angles but are not similar.
Does an enlargement change the angles of a shape?
No. An enlargement by scale factor k multiplies every length by k and leaves every angle unchanged, so the image is always similar to the object. When 0 < k < 1 the figure is reduced, but exam questions still call it an enlargement.
In the full PDF
- •Labelled diagrams of congruent quadrilaterals, similar triangles and overlapping triangles with intersecting cevians
- •Further worked examples: congruent quadrilaterals, choosing the correct congruence statement, an unknown on both parts of a ratio, and testing whether two quadrilaterals are similar
- •The butterfly case and the lamp-post shadow problem, solved by two methods
- •Enlargement and reduction examples, including a fractional scale factor, plus a map-scale preview with unit conversions
- •The remaining common mistakes and the four-step exam routine, with model answers
- •A one-page summary table comparing congruent and similar figures