Study Notes
Congruence and similarity involve comparing shapes to determine if they are identical in shape and size or just in shape. Congruence requires identical shape and size, while similarity requires identical shape but not necessarily size.
- 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 corresponding sides and the included angle of both triangles are equal. Example: Triangles with sides 5 cm, 4 cm, and angle 100°.
- ASA (Angle Side Angle) — Two corresponding angles and a corresponding included side of both triangles are equal. Example: Triangles with angles 60°, 30°, and side 5 cm.
- AAS (Angle Angle Side) — Two corresponding angles and a corresponding non-included side of both triangles are equal. Example: Triangles with angles 55°, 35°, and a non-included side equal.
- RHS (Right Angle Hypotenuse Side) — The corresponding hypotenuse and a corresponding side of both right-angled triangles are equal. Example: Right-angled triangles with hypotenuse and another side equal.
- Similarity — Two shapes are similar if they have the same shape but not necessarily the same size. Example: Triangles with angles 60° and proportional sides.
- AAA (Angle Angle Angle) — Two corresponding angles of both triangles are equal. Example: Triangles with two equal angles.
- Proportionality — The 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 length scale factor is 2, area is 4, and volume is 8.
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
Typical Exam Questions
- How do you prove two triangles are congruent using SSS? Check if all three sides are equal.
- What is the similarity test for triangles with two equal angles? Use AAA similarity.
- How do you find the ratio of volumes for similar solids? Cube the scale factor of the lengths.
What Examiners Usually Test
- Ability to identify and apply congruence tests
- Understanding of similarity and proportionality
- Calculating area and volume ratios for similar figures