Congruent figures match exactly. Similar figures have the same shape but may differ in size. Dilations make the difference.
Why it works
A dilation multiplies every length by the same factor $k$ and keeps every angle. Rigid motions then move the scaled copy into place without changing anything. So in similar figures, corresponding angles are equal and every pair of corresponding sides has the same ratio, $k$.
Key idea
Two figures are similar if a dilation together with rigid motions maps one onto the other. Then corresponding angles are equal and corresponding sides are proportional: $$\frac{\text{side in the image}}{\text{matching side in the original}} = k$$
Worked examples
Worked example: Missing side
$\triangle ABC \sim \triangle DEF$, with $AB = 6$, $DE = 9$ and $BC = 8$. Find $EF$.
- Scale factor from $ABC$ to $DEF$: $\frac{9}{6} = 1.5$.
- $EF = 1.5 \cdot 8 = 12$.
Answer: $12$
Worked example: Are they similar?
Is a 4 by 6 rectangle similar to a 6 by 9 rectangle? To a 5 by 7 rectangle?
- $\frac{6}{4} = 1.5$ and $\frac{9}{6} = 1.5$: the same factor, so yes.
- $\frac{5}{4} = 1.25$ but $\frac{7}{6} \approx 1.17$: different, so no.
Answer: yes; no
Worked example: Angles carry over
$\triangle ABC \sim \triangle DEF$ with $\angle A = 50^\circ$ and $\angle B = 60^\circ$. Find $\angle F$.
- Corresponding angles are equal: $\angle D = 50^\circ$, $\angle E = 60^\circ$.
- $\angle F = 180^\circ - 50^\circ - 60^\circ = 70^\circ$.
Answer: $70^\circ$
Common mistake
Adding instead of multiplying. Adding 2 to each side of a 4 by 6 rectangle gives 6 by 8, which is not similar: $\frac{6}{4} = 1.5$ but $\frac{8}{6} \approx 1.33$. Similar figures scale every length by the same factor.