Geometry - Similarity (of triangles, polygons)

Definition of Similarity: Two polygons are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional). Visually, similarity is like zooming in or out on a photograph; the shape remains identical, but the size changes. For triangles, if $\triangle ABC \sim \triangle PQR$, then $\angle A = \angle P$, $\angle B = \angle Q$, and $\angle C = \angle R$.

AA (Angle-Angle) Similarity Criterion: If two angles of one triangle are equal to two angles of another triangle, the two triangles are similar. This is the most common way to prove similarity. Visually, if you see two triangles with two matching corner angles, the third angle must also match because the sum of angles is $180^{\circ}$, making the triangles 'look' exactly the same in shape.

SAS (Side-Angle-Side) Similarity Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar. Visually, if you have two sides forming an identical angle, and those sides are scaled by the same factor (e.g., both doubled), the resulting triangles will be similar.

SSS (Side-Side-Side) Similarity Criterion: If the corresponding sides of two triangles are proportional, then their corresponding angles are equal and the triangles are similar. Visually, if you take a triangle and multiply every side length by a constant $k$, the new triangle will be a perfectly scaled version of the original.

Basic Proportionality Theorem (Thales's Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Visually, imagine a triangle $ABC$ with a line $DE$ drawn through it such that $DE \parallel BC$. This creates two segments on side $AB$ ($AD$ and $DB$) and two on side $AC$ ($AE$ and $EC$) such that $\frac{AD}{DB} = \frac{AE}{EC}$.

Ratio of Areas of Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If $\triangle ABC \sim \triangle PQR$, then $\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = \left(\frac{AB}{PQ}\right)^2 = \left(\frac{BC}{QR}\right)^2 = \left(\frac{AC}{PR}\right)^2$. Visually, if you double the sides of a triangle, the area becomes $2^2 = 4$ times larger.

Maps and Models (Scale Factor): For a map or a model with a scale factor $k = \frac{\text{Model Length}}{\text{Actual Length}}$, the following relationships hold: (i) $\text{Model Area} = k^2 \times \text{Actual Area}$ and (ii) $\text{Model Volume} = k^3 \times \text{Actual Volume}$. Visually, a $1:100$ scale model of a building will look tiny, but its surface area is reduced by $10,000$ and its volume/weight is reduced by $1,000,000$.

Internal Angle Bisector Theorem: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. Visually, if $AD$ is the bisector of $\angle A$ in $\triangle ABC$ meeting $BC$ at $D$, then $\frac{BD}{DC} = \frac{AB}{AC}$.