GCSE Maths: Congruent Triangles and Similar Shapes

Quantum Leap Star Ltd | GCSE Mathematics | Years 10–11 | Foundation and Higher

Learn how to recognise congruent triangles, match corresponding sides and angles, and calculate missing measurements in similar shapes. Start with the core methods, then try the Higher-tier extensions.

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Video lesson: 3 minutes 29 seconds. Pause when you need more time to follow a calculation.

Download the complete lesson notes (PDF), with original diagrams and worked examples

1. Congruent triangles: same shape and size

Congruent shapes fit exactly on top of each other. Corresponding sides have equal lengths and corresponding angles are equal. Translation, rotation and reflection preserve congruence.

• SSS: three corresponding sides are equal.
• SAS: two corresponding sides and the angle between them are equal.
• ASA / AAS: two corresponding angles and a corresponding side are equal.
• RHS: both triangles are right-angled, with equal hypotenuses and one other corresponding side.

Example: Two triangles each have sides 6 cm and 9 cm with a 40° angle between them. They are congruent by SAS.

Remember: AAA proves similarity, but does not prove congruence. SSA is not a general congruence test.

2. Similar shapes: matching angles and equal ratios

Similar shapes have the same shape, although their sizes may differ. Corresponding angles are equal, and corresponding lengths have the same ratio. Match sides using the vertices and angles, even when one shape has been turned or reflected.

Scale factor = new length ÷ matching original length.
New length = original length × scale factor.
Original length = new length ÷ scale factor.

Example: Similar rectangles have widths 5 cm and 15 cm. The scale factor from smaller to larger is 15 ÷ 5 = 3. If the smaller length is 8 cm, the larger length is 8 × 3 = 24 cm.

Triangles are similar by AA when two corresponding angles are equal. They are also similar when all three corresponding side ratios are equal, or when two corresponding side ratios and the included angles match.

3. Perimeters, areas and volumes

For length scale factor k, lengths and perimeters multiply by k. In the Higher-tier extension, areas multiply by k² and volumes multiply by k³.

Area example: Corresponding sides are 4 cm and 10 cm, giving k = 2.5. If the smaller area is 12 cm², the larger area is 12 × 2.5² = 75 cm².

Volume example: Similar solids have volume ratio 27 : 125, so their length ratio is 3 : 5. A 6 cm length on the smaller solid corresponds to 6 × 5 ÷ 3 = 10 cm.

4. Write clear mathematical reasons

When proving congruence or similarity, state each equality and give its reason. Then name the test. Use given information and angle facts rather than the appearance of a diagram. Convert measurements to the same units before calculating a ratio.

Read the full PDF for the RHS proof, shadow-height example and further Higher-tier methods. These lesson notes and diagrams were created for Quantum Leap Star Ltd.