A dilation alters lines; those not passing through the center become parallel lines, while those passing through remain the same. The dilation causes line segments to change based on the scale factor. Similarity between two figures is determined through similarity transformations - for triangles, this means equal angles and proportional sides. Properties of similarity transformations help establish the AA criterion for similar triangles. Triangle theorems involve proportional division by a parallel line and the Pythagorean Theorem applied to similarity.

Problem-solving and proof of relationships in geometric figures utilize triangle congruence and similarity criteria. Right-triangle side ratios, understood through similarity, are properties of the angles, thus bringing about definitions of trigonometric ratios for acute angles. The sine and cosine of complementary angles are related and used in calculations. Trigonometric ratios, alongside the Pythagorean theorem, are used for problem-solving in right triangles.

The area of a triangle is derived through the formula A = 1/2 ab sin(C), which can be achieved by drawing a line perpendicular to the opposite side from a vertex. The Laws of Sines and Cosines are also proven and applied. These laws can be applied to right and non-right triangles for the measurement of unknown quantities, in situations such as