What You'll Understand
- Congruence: Two shapes are congruent if they are identical in both shape and size. They can be rotated, reflected, or translated, but they're still the same.
- Similarity: Two shapes are similar if they have the same shape but different sizes. All corresponding angles are equal, and all corresponding sides are in the same ratio (proportional).
- Scale factors: The ratio between corresponding lengths in similar shapes. For areas and volumes, scale factors follow a power rule: if lengths scale by k, areas scale by k² and volumes by k³.
- Practical use: Identify congruent/similar shapes, find missing lengths/areas/volumes, and understand how 2D and 3D objects enlarge or reduce.
1. Congruence
What Does Congruent Mean?
Two shapes are congruent if they are identical in shape and size. Think of them as perfect copies.
A congruent shape may be:
- Reflected (flipped like a mirror image)
- Rotated (turned around)
- Translated (slid to a new position)
All of these are still congruent. What matters is that if you could physically move, flip, or turn one shape, it would fit exactly on top of the other with no gaps or overlaps.
Important: If one shape is an enlargement of another, they are not congruent — they are similar.
How to Prove Two Shapes Are Congruent
To convince an examiner that two shapes are congruent, you must show:
- All corresponding sides are the same length
- All corresponding angles are the same size
You don't need to show that they're facing in the same direction — rotation and reflection are allowed.
💡Examiner Tip: In exams, if the diagram is drawn to scale, you can use tracing paper. Trace one shape and see if it fits exactly on top of the other. But only use this if you're really unsure — showing your reasoning mathematically is always safer.
Worked Example: Identifying Congruent Shapes
Question:
Below are six shapes: A, B, C, D, E, and F. All are L-shaped. Which shapes are congruent to A?
(Imagine shape A is an L-shape. Shape C is the same L rotated. Shape D is the same L but reflected.)
2. Similarity
What Does Similar Mean?
Two shapes are similar if they have the same shape but different sizes. One is an enlargement of the other.
For shapes to be similar:
- All corresponding angles must be equal
- All corresponding sides must be in proportion (in the same ratio)
Key difference from congruence: Similar shapes can be different sizes. Congruent shapes must be the same size.
Proving Two Triangles Are Similar
For triangles, you only need to prove that the three corresponding angles are equal. If angles match, the sides will automatically be in proportion.
How to identify equal angles:
- Look for vertically opposite angles (they're always equal)
- Check for alternate angles on parallel lines (equal)
- angles in isosceles triangles
- given in the question