Similarity & Congruence

Welcome to Similarity & Congruence!

Two shapes are congruent if they are exactly the same — same shape, same size. Two shapes are similar if they have the same shape but can be different sizes. All corresponding angles are equal and all corresponding sides are in the same ratio. These ideas are fundamental to geometry and appear on every IGCSE 0580 paper.

Congruent: same shape AND same size | Similar: same shape, sides in ratio k | Area ∝ k² | Volume ∝ k³

Congruence

SSS, SAS, ASA, AAS, RHS

Similarity Tests

AA, SAS, SSS for triangles

Scale Factor

Finding missing lengths

Area & Volume

k², k³ scale factors

Maps & Scale Drawings

Real-world applications

Proof Technique

Exam-style proofs

1. Congruent Figures

Two figures are congruent if one can be mapped exactly onto the other by a combination of reflections, rotations and translations (no stretching or enlargement). Every corresponding side has the same length and every corresponding angle has the same measure.

Congruent shapes: all corresponding sides equal AND all corresponding angles equal

Symbol: Triangle ABC ≅ Triangle DEF means the two triangles are congruent.
Order matters: If ABC ≅ DEF, then AB = DE, BC = EF, CA = FD, and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

When you write a congruence statement, always list vertices in matching order. This tells the reader which sides and angles correspond.

2. The Five Congruence Conditions

You only need to check enough information to guarantee congruence. There are five accepted conditions for triangles.

SSSAll three sides of one triangle equal the three sides of the other (Side–Side–Side).

SASTwo sides and the included angle (the angle between those two sides) are equal (Side–Angle–Side).

ASATwo angles and the included side (the side between those two angles) are equal (Angle–Side–Angle).

AASTwo angles and a non-included side are equal. (Because if two angles are equal, the third must be too.) (Angle–Angle–Side).

RHSRight angle, Hypotenuse, and one other Side are equal. Only for right-angled triangles.

SSA is NOT a valid congruence condition (two sides and a non-included angle can give two different triangles). Neither is AAA alone (it only proves similarity).

3. Similar Figures

Two figures are similar if all corresponding angles are equal and all corresponding sides are in the same ratio. One is an enlargement of the other.

Similar shapes: all corresponding angles equal AND corresponding sides in ratio k (the scale factor)

Symbol: Triangle ABC ∼ Triangle DEF
Scale factor k: k = (length in larger shape) ÷ (corresponding length in smaller shape)
If k > 1, the second shape is larger. If k < 1, the second shape is smaller. If k = 1, the shapes are congruent.

Finding a missing side: If AB = 6, DE = 9, BC = 8, find EF.
Scale factor k = DE/AB = 9/6 = 1.5
EF = BC × k = 8 × 1.5 = 12

Always identify which triangles are similar FIRST and write them in matching order. Then set up the ratio using two known corresponding sides to find k.

4. The Three Similarity Tests for Triangles

AA (Angle–Angle)If two angles of one triangle equal two angles of another, the triangles are similar. (The third angles must also be equal since angles sum to 180°.)

SAS SimilarityTwo pairs of corresponding sides are in the same ratio AND the included angles are equal.

SSS SimilarityAll three pairs of corresponding sides are in the same ratio (k = a₁/a₂ = b₁/b₂ = c₁/c₂).

AA is by far the most common similarity test in IGCSE exams. Parallel lines, common angles, and vertically opposite angles are the most frequent sources of equal angles.

5. Scale Factor: Length, Area and Volume

When two shapes are similar with linear scale factor k, measurements in two and three dimensions scale differently.

Length scale factor = k | Area scale factor = k² | Volume scale factor = k³

Example: Two similar cylinders have radii 4 cm and 10 cm.
Linear scale factor k = 10 ÷ 4 = 2.5
Area scale factor = 2.5² = 6.25 ⟹ curved surface area × 6.25
Volume scale factor = 2.5³ = 15.625 ⟹ volume × 15.625

Working backwards from area: Two similar triangles have areas 25 cm² and 100 cm².
Area ratio = 100/25 = 4 = k² ⟹ k = √4 = 2
So the linear scale factor is 2.

Remember: area involves two dimensions, so it scales by k². Volume involves three dimensions, so it scales by k³. Never confuse these with the linear scale factor k.

6. Maps and Scale Drawings

A map or scale drawing is similar to the real object. The scale is the ratio of a length on the map to the corresponding real-world length.

Map scale = map length : real length e.g. 1 : 50 000 means 1 cm on map = 50 000 cm = 500 m in reality

Example 1: A map has scale 1 : 25 000. Two towns are 8 cm apart on the map. What is the real distance?
Real distance = 8 × 25 000 = 200 000 cm = 2 km

Example 2: A scale drawing of a room uses scale 1 : 40. The room is 5 m × 4 m in reality. What are the dimensions on the drawing?
5 m = 500 cm ⟹ 500 ÷ 40 = 12.5 cm
4 m = 400 cm ⟹ 400 ÷ 40 = 10 cm

7. Similar Shapes in 3D

The k², k³ rules apply to all similar 3D shapes — cylinders, cones, spheres, pyramids, cuboids — as long as they are geometrically similar (all corresponding lengths in ratio k).

Similar cones: Heights 6 cm and 15 cm.
k = 15/6 = 2.5 | Area factor = 6.25 | Volume factor = 15.625
If smaller cone has volume 48 cm³, larger cone has volume 48 × 15.625 = 750 cm³.

Always identify the linear scale factor first, then square it for area and cube it for volume. Watch for questions that give you an area or volume ratio and ask for a length — reverse the process using square root or cube root.

8. Proving Triangles Similar or Congruent (Exam Technique)

Proof questions require a precise, structured argument. For IGCSE, state which angles (or sides) are equal and why, then quote the condition you are using.

Structure for a similarity proof:
1. Identify the angle equality and give the geometric reason (e.g. "alternate angles, AB ∥ CD").
2. Identify a second angle equality (e.g. "common angle at X").
3. Conclude: "Therefore triangles ABC and XYZ are similar (AA)."

Structure for a congruence proof:
1. State the first pair of equal sides/angles with reason.
2. State the second pair with reason.
3. State the third piece of information with reason.
4. Conclude: "Therefore triangles ABC and DEF are congruent (SAS)."

Common reasons to use: common side, common angle, vertically opposite angles, alternate angles (parallel lines), corresponding angles (parallel lines), base angles of isosceles triangle, given information.

Example 1 — Proving Congruence (SAS)

Setup: In triangles ABC and DEF: AB = DE = 7 cm, BC = EF = 5 cm, ∠ABC = ∠DEF = 65°. Prove the triangles are congruent.

Step 1: AB = DE = 7 cm (given) — one pair of equal sides.

Step 2: BC = EF = 5 cm (given) — second pair of equal sides.

Step 3: ∠ABC = ∠DEF = 65° (given) — the included angle between the two sides is equal.

Conclusion: Triangle ABC ≅ Triangle DEF (SAS — two sides and the included angle are equal).

Example 2 — Proving Similarity (AA)

Setup: AB is parallel to DE, and the triangles ABC and DEC share vertex C. Prove triangles ABC and DCE are similar.

Step 1: ∠BAC = ∠EDC (alternate angles, AB ∥ DE).

Step 2: ∠ACB = ∠DCE (vertically opposite angles).

Conclusion: Triangles ABC and DCE are similar (AA — two pairs of angles are equal).

Note: Since two angles are equal the third must be too (angles in a triangle = 180°), but two is sufficient for AA.

Example 3 — Finding a Missing Length

Setup: Triangles PQR and XYZ are similar with PQ = 8 cm, QR = 12 cm, XY = 10 cm. Find YZ.

Step 1: Identify the scale factor: k = XY / PQ = 10 / 8 = 1.25.

Step 2: Apply the scale factor to find YZ: YZ = QR × k = 12 × 1.25 = 15 cm.

Check: All sides of XYZ should be 1.25× the corresponding sides of PQR. ✓

Example 4 — Area Scale Factor

Setup: Two similar rectangles have lengths 5 cm and 15 cm. The smaller rectangle has area 20 cm². Find the area of the larger rectangle.

Step 1: Linear scale factor k = 15 / 5 = 3.

Step 2: Area scale factor = k² = 3² = 9.

Step 3: Area of larger rectangle = 20 × 9 = 180 cm².

Example 5 — Volume Scale Factor (Reverse)

Setup: Two similar cones have volumes 54 cm³ and 128 cm³. The smaller cone has height 9 cm. Find the height of the larger cone.

Step 1: Volume ratio = 128 / 54 = 64/27.

Step 2: Volume scale factor = k³, so k³ = 64/27 ⟹ k = ∛(64/27) = 4/3.

Step 3: Height of larger cone = 9 × (4/3) = 12 cm.

Example 6 — Map Scale

Setup: A map has scale 1 : 50 000. Two villages are 6.4 cm apart on the map. What is the actual distance in km?

Step 1: Actual distance = 6.4 × 50 000 = 320 000 cm.

Step 2: Convert to km: 320 000 ÷ 100 000 = 3.2 km.

Similar Triangles — Interactive Visualiser

Adjust the scale factor k to see how the second triangle changes. Corresponding sides are colour-coded. The ratio of each pair of sides is always k.

Scale factor k = k = 1.50

Drag the slider to change k.

Area and Volume Scale Factor Calculator

Enter a linear scale factor to see the corresponding area and volume scale factors.

k =

Enter k and click Calculate.

Exercise 1 — Congruence Conditions

Each question describes two triangles. Enter the numeric code for the congruence condition: 1 = SSS, 2 = SAS, 3 = ASA, 4 = AAS, 5 = RHS, 0 = Not congruent.

1. Triangle A has sides 5, 7, 9 cm. Triangle B has sides 5, 7, 9 cm. Which condition? (SSS=1, SAS=2, ASA=3, AAS=4, RHS=5)

2. Two triangles share: two sides of 6 cm and 8 cm, and the angle between them is 50° in both. Which condition?

3. Two triangles share: angles 40° and 70°, with the side between them = 10 cm in both. Which condition?

4. Two right-angled triangles share: hypotenuse 13 cm and one leg 5 cm. Which condition?

5. Two triangles share: angles 35° and 85°, and one side NOT between those angles = 7 cm in both. Which condition?

6. Triangle C has sides 4, 6 cm and angle 30° (not between those sides). Triangle D has the same information. Is this SSA? Enter 0 for not congruent.

7. Two triangles share: one side of 11 cm, one angle of 60°, and another angle of 40° (angles not adjacent to the side). Which condition?

8. Two right-angled triangles have hypotenuse 17 cm and a leg of 8 cm each. Which condition?

9. Triangles with all three angles equal but different sizes. Enter 0 (not congruent, only similar).

10. Two triangles: sides 3, 4, 5 and 3, 4, 5 cm. Which condition?

Exercise 2 — Missing Lengths in Similar Triangles

Use the scale factor to find the missing length. Give your answer to 1 decimal place where necessary.

1. Triangles ABC ∼ DEF. AB = 4 cm, DE = 6 cm, BC = 5 cm. Find EF.

2. Triangles PQR ∼ XYZ. PQ = 8 cm, XY = 12 cm, QR = 6 cm. Find YZ.

3. Triangles ABC ∼ DEF. AB = 10 cm, DE = 15 cm, AC = 8 cm. Find DF.

4. Triangles LMN ∼ PQR. LM = 6 cm, PQ = 9 cm, MN = 10 cm. Find QR.

5. Triangles ABC ∼ DEF. AB = 5 cm, DE = 20 cm, BC = 7 cm. Find EF.

6. Triangles PQR ∼ XYZ. PQ = 9 cm, XY = 6 cm, QR = 12 cm. Find YZ (smaller triangle).

7. Similar triangles: small side 3 cm corresponds to large side 7.5 cm. Another small side is 4 cm. Find the corresponding large side.

8. Triangles ABC ∼ DEF. AB = 14, DE = 7, BC = 11. Find EF (round to 1 d.p.).

9. Two similar triangles: corresponding sides are x and 18, with scale factor k = 3. Find x.

10. Triangles ABC ∼ DEF. BC = 8, EF = 5, AC = 12. Find DF (round to 1 d.p.).

Exercise 3 — Area Scale Factor (k²)

Use the area scale factor k² to find missing areas. Give integer answers or 1 d.p. where needed.

1. Two similar triangles have corresponding sides 3 cm and 6 cm. What is the area scale factor?

2. Two similar shapes have sides 4 cm and 12 cm. The smaller has area 8 cm². Find the larger area.

3. Similar rectangles with sides 5 cm and 10 cm. Smaller area = 30 cm². Find larger area.

4. Two similar triangles have areas 9 cm² and 36 cm². Find the linear scale factor k.

5. Two similar triangles have areas 16 cm² and 100 cm². Find k.

6. Two similar shapes: sides 6 cm and 9 cm. Larger area = 243 cm². Find smaller area.

7. Similar pentagons: sides 2 cm and 7 cm. Smaller area = 4 cm². Find the larger area.

8. Two similar triangles have areas 50 cm² and 200 cm². Find k.

9. Two similar shapes: k = 5. Smaller area = 7 cm². Find larger area.

10. Two similar triangles have areas 18 cm² and 32 cm². A side of the smaller triangle is 6 cm. Find the corresponding side of the larger. (Round to 1 d.p.)

Exercise 4 — Volume Scale Factor (k³)

Apply k³ to find missing volumes. Round to 1 d.p. where necessary.

1. Two similar cylinders have heights 2 cm and 6 cm. What is the volume scale factor?

2. Two similar cones have heights 3 cm and 9 cm. Smaller volume = 10 cm³. Find larger volume.

3. Two similar cuboids have sides 4 cm and 8 cm. Smaller volume = 48 cm³. Find larger volume.

4. Two similar spheres have radii 2 cm and 6 cm. What is the volume scale factor?

5. Two similar solids have volumes 8 cm³ and 64 cm³. Find the linear scale factor k.

6. Two similar solids have volumes 27 cm³ and 125 cm³. Find k (as a fraction — enter the decimal).

7. Two similar pyramids have volumes 2 cm³ and 54 cm³. Find k.

8. Two similar prisms: k = 4. Smaller volume = 5 cm³. Find larger volume.

9. Two similar cones have volumes 32 cm³ and 4 cm³. Find k (larger to smaller).

10. Two similar solids have volumes 250 cm³ and 2000 cm³. Find k.

Exercise 5 — Maps and Scale Drawings

Map scale questions. Read carefully — some ask for map lengths (in cm), others for real distances (in km or m).

1. Scale 1 : 100. A wall is 5 m long. Length on drawing in cm?

2. Scale 1 : 50 000. Two towns are 4 cm apart on a map. Real distance in km?

3. Scale 1 : 25 000. A forest is 8 cm wide on a map. Real width in km?

4. Scale 1 : 200. A room is 6 m × 4 m. Length of longer side on plan in cm?

5. Scale 1 : 50 000. A road is 12.5 km long. Length on map in cm?

6. Scale 1 : 1000. A garden is 3 cm × 5 cm on a plan. Real length of longer side in metres?

7. Scale 1 : 10 000. Two points are 7.5 cm apart on a map. Real distance in km?

8. Scale 1 : 500. A field is 150 m long. Length on drawing in cm?

9. A map shows 1 cm = 5 km. The map distance between two cities is 3.5 cm. Real distance in km?

10. Scale 1 : 20 000. A lake is 3 km long. Length on map in cm?

Practice — 20 Mixed Questions

Mixed practice covering congruence, similarity, scale factor, area, volume and maps. Round to 1 d.p. where necessary.

1. Two similar triangles: sides 4 cm and 10 cm. What is the linear scale factor k? (Enter as decimal)

2. Triangles ABC ∼ DEF. AB = 6, DE = 9, BC = 8. Find EF.

3. Two similar shapes: sides 5 cm and 15 cm. Find the area scale factor.

4. Two similar cylinders: heights 4 cm and 12 cm. Find the volume scale factor.

5. Scale 1 : 50 000. Map distance = 6 cm. Real distance in km?

6. Two similar triangles have areas 25 cm² and 100 cm². Find k.

7. Two similar solids have volumes 8 cm³ and 216 cm³. Find k.

8. Triangles ABC ∼ DEF. AB = 5, DE = 20, BC = 9. Find EF.

9. Similar shapes: k = 3. Smaller area = 12 cm². Find larger area.

10. Similar shapes: k = 4. Smaller volume = 6 cm³. Find larger volume.

11. Two similar rectangles have areas 50 cm² and 200 cm². Find k.

12. Scale 1 : 25 000. Two points are 8 cm apart on a map. Real distance in km?

13. Two similar cones: heights 5 cm and 10 cm. Smaller volume = 40 cm³. Find larger volume.

14. Triangles ABC ∼ DEF. BC = 9, EF = 12, AC = 6. Find DF (1 d.p.).

15. Two similar solids have volumes 54 cm³ and 250 cm³. Find k to 2 d.p. (Enter 2 d.p.)

16. Two similar triangles have areas 4 cm² and 9 cm². A side of the larger is 15 cm. Find corresponding side of smaller.

17. Scale 1 : 500. A corridor is 150 m long. Length on plan in cm?

18. Two similar shapes: sides 8 cm and 20 cm. Larger area = 500 cm². Find smaller area (in cm²).

19. Congruence code: Two triangles — all three sides equal. Enter 1 (SSS).

20. Triangles ABC ∼ DEF. AB = 7, DE = 14, BC = 5. Find EF.

Challenge — 8 Harder Questions

These questions combine similarity with other topics. Round to 1 d.p. unless told otherwise.

1. A triangle has base 12 cm and height 8 cm (area = 48 cm²). A similar triangle has base 18 cm. Find the area of the larger triangle in cm².

2. Two similar spheres have surface areas 36π cm² and 144π cm². Find the ratio of their volumes. Give the ratio as a single number (larger ÷ smaller, exact integer).

3. In triangle ABC, DE is parallel to BC with D on AB and E on AC. AD = 4 cm, DB = 6 cm, BC = 15 cm. Find DE in cm.

4. Two similar cones have total surface areas 48π cm² and 108π cm². The smaller cone has volume 32π cm³. Find the volume of the larger cone in terms of π (give just the coefficient).

5. A scale model of a building uses scale 1 : 200. The model has a floor area of 0.3 m². What is the real floor area in m²?

6. Triangle ABC is similar to triangle ADE where D is on AB and E is on AC. AB = 15 cm, AD = 5 cm, BC = 12 cm. Find DE in cm.

7. Two similar solids have surface areas 75 cm² and 300 cm². The larger solid has volume 640 cm³. Find the volume of the smaller solid in cm³.

8. A trapezium ABCD has AB ∥ CD. Diagonals AC and BD intersect at X. AB = 9 cm, CD = 6 cm, BX = 12 cm. Using similar triangles, find XD in cm. (Round to 1 d.p.)