Definition of Similar Triangles
📐 What Are Similar Triangles?
Two triangles are similar if they have the same shape but not necessarily the same size.
Formal Definition
Two triangles $\triangle ABC$ and $\triangle DEF$ are similar (written $\triangle ABC \sim \triangle DEF$) if and only if:
- All corresponding angles are equal (congruent):
$\angle A = \angle D$, $\quad \angle B = \angle E$, $\quad \angle C = \angle F$ - All corresponding sides are proportional:
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$
The constant $k$ is called the scale factor (or ratio of similarity).
🔑 Both Conditions Must Hold
For two triangles to be similar, both conditions must be satisfied simultaneously:
| Condition | What It Means | Think of It As... |
|---|---|---|
| Equal Angles | $\angle A = \angle D$, $\angle B = \angle E$, $\angle C = \angle F$ | Same shape |
| Proportional Sides | $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ | Scaled version (bigger or smaller) |
⚠️ Important Note: For triangles specifically, if one condition is true, the other is automatically true as well. This is a special property of triangles that does NOT hold for other polygons!
📏 The Scale Factor
The scale factor $k$ tells us how much one triangle has been enlarged or reduced compared to the other.
- If $k > 1$: The second triangle is smaller than the first (we divided by a smaller number)
- If $k = 1$: The triangles are congruent (same size and shape)
- If $0 < k < 1$: The second triangle is larger than the first