Definition of Similarity
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional:
\[ \triangle ABC \sim \triangle DEF \quad \text{if and only if} \quad \begin{cases} \angle A = \angle D, \\ \angle B = \angle E, \\ \angle C = \angle F, \\ \dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF} \end{cases} \]Where \(\sim\) is the symbol of similarity of two triangles.
Similarity Postulates
1. Angle-Angle (AA) Similarity
If two angles of one triangle are equal to two angles of another triangle, the triangles are similar since the third angle automatically follows from the angle sum property of triangles.
Example:
Given \(\angle A = \angle D\) and \(\angle B = \angle E\),
\(\implies \triangle ABC \sim \triangle DEF\)
2. Side-Angle-Side (SAS) Similarity
If two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.
Example:
\(\dfrac{AB}{DE} = \dfrac{AC}{DF}\) and \(\angle A = \angle D\)
\(\implies \triangle ABC \sim \triangle DEF\)
3. Side-Side-Side (SSS) Similarity
If all three sides of one triangle are proportional to the corresponding sides of another triangle, the triangles are similar.
Example:
\(\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF}\)
\(\implies \triangle ABC \sim \triangle DEF\)
Key Theorems
Basic Proportionality Theorem (Thales' Theorem)
If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
Example:
If \(DE \parallel BC\) in \(\triangle ABC\),
then \(\dfrac{AD}{DB} = \dfrac{AE}{EC}\).
Right Triangle Similarity
The altitude to the hypotenuse of a right triangle creates two smaller triangles similar to each other and to the original triangle.
Example:
In right \(\triangle ABC\) with altitude \(CD\):
\(\triangle ABC \sim \triangle ACD \sim \triangle CBD\) (all three triangles are similar to each other).
Midline Theorem
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
Example:
If \(D\) and \(E\) are midpoints of \(AB\) and \(AC\),
then \(DE \parallel BC\) and \(DE = \dfrac{1}{2}BC\).
Important Properties
Ratio of Areas
The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides.
\[ \dfrac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\dfrac{AB}{DE}\right)^2 \]Angle Bisector Theorem
An angle bisector divides the opposite side into segments proportional to the adjacent sides.
Example:
If \(AD\) bisects \(\angle A\) in \(\triangle ABC\),
then \(\dfrac{BD}{DC} = \dfrac{AB}{AC}\).