Similarity describes shapes that have the same form but may differ in size. Two shapes are similar when their corresponding sides are in proportion and corresponding angles are equal. For triangles, equal angles automatically imply proportional sides.
1. Definition of Similar Shapes
Two shapes are similar if they have the same shape and their corresponding sides are in proportion.
One shape is an enlargement of the other
Similarity does not imply congruence (similar shapes can be different sizes)
Two similar shapes could be different enlargements of each other
Example: Two rectangles with sides 15 cm × 5 cm and 6 cm × 2 cm are similar because \frac{15}{6} = \frac{5}{2} = 2.5
2. Proving Triangles Similar
To show two triangles are similar, prove that all three corresponding angles are equal.
If angles are equal, corresponding lengths are automatically in proportion
Use angle properties: isosceles triangles, vertically opposite angles, alternate/corresponding angles on parallel lines
For each pair of corresponding angles: state they are equal and give a reason
Hourglass formation: Similar triangles often appear opposite each other when two lines cross. Look for vertically opposite equal angles.
3. Proving Non-Triangular Shapes Similar
For polygons other than triangles, show that corresponding sides are in proportion.
Identify corresponding sides
Find the scale factor: divide one side by its corresponding side
Check the scale factor is the same for all pairs of corresponding sides
If scale factor is constant, the shapes are similar.
4. Scale Factor
The scale factor (k) links corresponding lengths:
\[ \frac{\text{length on shape 2}}{\text{corresponding length on shape 1}} = k \]
(k > 1): second shape is bigger
(0 < k < 1): second shape is smaller
5. Worked Example: Proving Rectangles Similar
6. Worked Example: Triangles with Parallel Lines
Prove rectangles with dimensions 15 cm × 5 cm and 6 cm × 2 cm are similar.
Solution:
Scale factor from lengths: \frac{15}{6} = 2.5
Scale factor from widths: \frac{5}{2} = 2.5
Same scale factor for both pairs \Rightarrow rectangles are similar
When AB and CD are parallel, triangles ABX and CDX are similar.
Reasons:
(\angle AXB = \angle CXD) (vertically opposite)
(\angle ABC = \angle BCD) (alternate angles on parallel lines)
(\angle BAD = \angle ADC) (alternate angles on parallel lines)
All three corresponding angles equal \Rightarrow triangles similar.
