Applying Similarity to Solving Problems
What is Similarity?
Two figures are similar (denoted by ~) if they have the same shape but not necessarily the same size. For triangles, this means:
Condition 1: Corresponding Angles are Equal
If $\triangle ABC \sim \triangle DEF$, then:
$$\angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F$$
Condition 2: Corresponding Sides are Proportional
If $\triangle ABC \sim \triangle DEF$, then:
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$
where $k$ is the scale factor.
Similarity Criteria (How to Prove Triangles are Similar)
AA (Angle-Angle)
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
This is the most commonly used criterion!
SSS (Side-Side-Side)
If all three pairs of corresponding sides are proportional, the triangles are similar.
$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$$
SAS (Side-Angle-Side)
If two pairs of sides are proportional and the included angles are equal, the triangles are similar.