Summary
Congruence and similarity involve comparing shapes to determine if they are identical in shape and size or just in shape, respectively.
- Congruence — Two shapes are congruent if they have the same shape and size. Example: Triangles with sides 4.5 cm, 5 cm, and 4 cm are congruent by SSS.
- SSS (Side Side Side) — The three corresponding sides of two triangles have the same length. Example: Triangles with sides 4.5 cm, 5 cm, and 4 cm.
- SAS (Side Angle Side) — Two sides and the included angle of two triangles are equal. Example: Triangles with sides 5 cm, 4 cm, and an angle of 100°.
- ASA (Angle Side Angle) — Two angles and the included side of two triangles are equal. Example: Triangles with angles 60°, 30°, and a side of 5 cm.
- AAS (Angle Angle Side) — Two angles and a non-included side of two triangles are equal. Example: Triangles with angles 55°, 35°, and a non-included side equal.
- RHS (Right Angle Hypotenuse Side) — The hypotenuse and one side of right-angled triangles are equal. Example: Right-angled triangles with equal hypotenuse and one side.
- Similarity — Two shapes are similar if they have the same shape but not necessarily the same size. Example: Triangles with two equal angles (AAA).
- AAA (Angle Angle Angle) — Two corresponding angles of two triangles are equal. Example: Triangles with two equal angles.
- Proportionality — Corresponding sides of two triangles are proportional. Example: Triangles with sides in the ratio 1:2.
- Ratio of Area/Volume — The ratio of areas is the square of the scale factor, and the ratio of volumes is the cube of the scale factor. Example: If the scale factor is 2, area ratio is 2², volume ratio is 2³.
Exam Tips
Key Definitions to Remember
- Congruence: Same shape and size.
- Similarity: Same shape, different size.
- SSS, SAS, ASA, AAS, RHS for congruence.
- AAA for similarity.
Common Confusions
- Confusing congruence with similarity.
- Misidentifying included and non-included sides in ASA and AAS.
Typical Exam Questions
- How do you prove two triangles are congruent? Use SSS, SAS, ASA, AAS, or RHS criteria.
- How do you prove two triangles are similar? Use AAA or proportionality of sides.
- What is the ratio of areas if the scale factor is 3? The ratio is 3² = 9.
What Examiners Usually Test
- Understanding and application of congruence tests.
- Understanding and application of similarity tests.
- Calculating ratios of areas and volumes using scale factors.