Congruence and similarity
Congruent triangles
shapes are exactly the same shape and size (one may be turned or flipped). Two triangles are congruent if one of these is true:
| Condition | Meaning |
|---|---|
| all three sides equal | |
| two sides and the angle between them equal | |
| two angles and a corresponding side equal | |
| right angle, hypotenuse and one other side equal |
: triangles with the same three angles can be different sizes.
Example: triangles P and Q both have sides of 5 cm and 7 cm, with an angle of 40° between them. Are they congruent?
- Two sides and the angle between them are equal
- Yes, by SAS
Exam tip
Name the condition and say which sides and angles are equal, not just "they look the same".
Key terms
Similar shapes
shapes are the same shape but can be different sizes. Their angles are equal and their sides are in the same ratio. The k is how many times bigger one is.
- Find a pair of matching sides.
- Scale factor = new length ÷ original length.
- Multiply (or divide) the other lengths by the scale factor.
Example: a triangle has sides of 4 cm, 6 cm and 8 cm. A similar triangle has a longest side of 12 cm. Find its other sides.
- Matching sides: 8 cm and 12 cm
- k = 12 ÷ 8 = 1.5
- 4 × 1.5 = 6 and 6 × 1.5 = 9
- 6 cm and 9 cm
Key terms
Similar triangles inside each other
When a line inside a triangle is parallel to one side, it cuts off a smaller triangle that is similar to the big one. The angles match because of corresponding angles.
- Separate the two triangles, and match up their sides.
- Find the scale factor from a pair of matching sides.
- Use it to find the missing side.
Example: in triangle ABC, D is on AB and E is on AC, with DE parallel to BC. AD = 4 cm, AB = 10 cm and DE = 6 cm. Find BC.
- Triangle ADE is similar to triangle ABC
- Matching sides AD and AB: k = 10 ÷ 4 = 2.5
- BC = 6 × 2.5 = 15 cm
Common mistake
Use the whole side of the big triangle, AB = 10 cm, not DB = 6 cm.
Area and volume scale factors (Higher tier only)
If the lengths of similar shapes are in the ratio 1 : k:
- the areas are in the ratio 1 : k²
- the volumes are in the ratio 1 : k³
Example: two similar cylinders have heights of 6 cm and 9 cm. The smaller one has a volume of 160 cm³ and a surface area of 96 cm². Find the volume and surface area of the larger one.
- Length scale factor: k = 9 ÷ 6 = 1.5
- Volume: 160 × 1.5³ = 160 × 3.375 = 540 cm³
- Area: 96 × 1.5² = 96 × 2.25 = 216 cm²
Example: two similar shapes have areas of 20 cm² and 45 cm². Find the length scale factor.
- Area scale factor: 45 ÷ 20 = 2.25 = k²
- k = √2.25 = 1.5