Same Shape, Same or Different Size: Congruence & Similarity
Congruent figures are identical in shape and size — one can be placed exactly on top of the other through rigid transformations. Similar figures have the same shape but may differ in size, with all corresponding angles equal and corresponding sides in proportion. These concepts underpin proofs, scale drawings, and real-world applications like map reading and model building.
Components of Congruence & Similarity
This section covers the key principles:
- Congruence Criteria: Triangles are congruent if they satisfy SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), or AAS (angle-angle-side).
- Similarity Criteria: Triangles are similar if they satisfy AA (angle-angle), SSS similarity (all sides proportional), or SAS similarity (two sides proportional with included angle equal).
- Corresponding Parts: In congruent or similar figures, matching sides and angles are called corresponding parts. In congruent figures, corresponding parts are equal (CPCTC).
- Scale Factor: The ratio of corresponding side lengths in similar figures. If the scale factor is k, then areas scale by k² and volumes scale by k³.
Examples of Congruence & Similarity
Congruence Criteria Examples
- Two triangles have sides 5, 7, 9 cm and 5, 7, 9 cm. By SSS, they are congruent.
- Triangle A has sides 6, 8 with an included angle of 50°. Triangle B has the same. By SAS, they are congruent.
- Triangle P has angles 40° and 60° with the side between them measuring 10 cm. Triangle Q matches. By ASA, they are congruent.
Similarity Criteria Examples
- Triangle X has angles 30° and 70°. Triangle Y has angles 30° and 70°. By AA, they are similar (the third angle must be 80° in both).
- Triangle A has sides 3, 4, 5. Triangle B has sides 6, 8, 10. Each pair is in ratio 1:2, so by SSS similarity, they are similar.
- Two triangles share an angle of 45°, and the sides forming that angle are in ratio 2:3 in both. By SAS similarity, they are similar.
Corresponding Parts Examples
- Triangles ABC and DEF are congruent. If AB = 12, then DE = 12. If angle A = 55°, then angle D = 55°.
- In similar triangles with a scale factor of 3, if one side of the smaller triangle is 4 cm, the corresponding side of the larger is 4 × 3 = 12 cm.
- Two congruent rectangles have matching diagonals, widths, and lengths — every measurement is identical.
Scale Factor Examples
- A model car is built at a 1:24 scale. If the real car is 4.8 meters long, the model is 4.8 ÷ 24 = 0.2 meters (20 cm).
- Two similar triangles have sides in ratio 2:5. If the smaller triangle has area 12 cm², the larger has area 12 × (5/2)² = 12 × 6.25 = 75 cm².
- A map has a scale of 1:50,000. A distance of 3 cm on the map represents 3 × 50,000 = 150,000 cm = 1.5 km in real life.