Similarity

What is Similarity?

Similar shapes are shapes that have exactly the same shape but are different sizes. Think of them as photocopies of each other – one might be enlarged or reduced, but they look identical otherwise.

For two shapes to be similar:

• All their corresponding angles must be equal (corresponding means matching angles in the same position)
• All their corresponding sides must be in proportion (in proportion means they are in the same ratio)
• One shape is an enlargement or reduction of the other

Example: A small photograph and a poster-sized version of the same photograph are similar shapes – same image, different sizes.

Proving That Two Shapes Are Similar

Proving Triangles Are Similar

To prove that two triangles are similar, you need to show that all three pairs of corresponding angles are equal.

This is sometimes called the AAA property (Angle-Angle-Angle). Once you've shown the angles are equal, the sides will automatically be in proportion – you don't need to check the sides separately for triangles.

How to find equal angles:

Use geometric properties to identify equal angles:

• Isosceles triangles have two equal angles
• Vertically opposite angles are equal (angles formed when two lines cross)
• Alternate angles on parallel lines are equal
• Corresponding angles on parallel lines are equal

Step-by-step approach for triangles:

1. Identify each pair of corresponding angles
2. For each pair, state that they are equal
3. Give a clear geometric reason why they are equal (e.g., "vertically opposite angles" or "alternate angles on parallel lines")
4. Once all three pairs of angles are shown to be equal, conclude that the triangles are similar

Example: If you have two triangles where all three angles in the first triangle are 60°, 70°, and 50°, and all three angles in the second triangle are also 60°, 70°, and 50°, then the triangles are similar.

Proving Other Shapes Are Similar

For shapes that aren't triangles (like rectangles, quadrilaterals, pentagons, etc.), you need to show that all corresponding sides are in the same ratio.

Step-by-step approach:

1. Take one pair of corresponding sides
2. Divide the length from the second shape by the length from the first shape – this gives you the scale factor
3. Repeat this for all other pairs of corresponding sides
4. If the scale factor is the same for every pair of sides, the shapes are similar

Example: Two rectangles with dimensions 6 cm × 4 cm and 15 cm × 10 cm.

• Scale factor for lengths: 15 ÷ 6 = 2.5
• Scale factor for widths: 10 ÷ 4 = 2.5
• Both scale factors are 2.5, so the rectangles are similar