Two triangles are similar if:

In geometry, two polygons are **similar** if they have the same shape but not necessarily the same size. For two triangles to be similar, they must satisfy two fundamental conditions simultaneously: 1. **Corresponding Angles are Equal:** The internal angles of one triangle must be exactly equal to the corresponding internal angles of the other triangle. For example, if $\triangle ABC \sim \triangle DEF$, then $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$. 2. **Corresponding Sides are Proportional:** The ratio of the lengths of their corresponding sides must be constant (this constant is called the scale factor, $k$). Mathematically, for $\triangle ABC$ and $\triangle DEF$ to be similar: $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$ While **congruent** triangles are always similar (with a ratio of $1:1$), similar triangles are not necessarily congruent. Having the same perimeter or area does not guarantee similarity, as the shape (angles) can vary significantly while keeping those values constant.