Short Answer 
Ratios compare two quantities, typically in the format a:b, and are essential for understanding relationships between objects. In geometry, two triangles are similar if they have two equal angles, which leads to proportional corresponding sides, allowing for the derivation of ratios from these triangles by calculating their side lengths.
Step 1: Understand Ratios
A ratio compares two quantities, illustrating how many times one value contains or is contained within the other. The format for a ratio is generally expressed as *a:b* or *b:a*. It’s important to maintain the same units for both quantities to ensure an accurate comparison, and the ratio inherently has no units, making it a pure number.
Step 2: Applying Angle-Angle Similarity
Two triangles can be called similar if they have two corresponding angles of equal measure. This similarity implies that the third angle will also be equal because the total degree measure of a triangle is fixed. Therefore, studying the angles helps to establish the similarity of triangles and ultimately their proportional sides.
Step 3: Deriving Ratios from Similar Triangles
For similar triangles, if angles are proven equal, then the ratios of the lengths of corresponding sides will also hold. Given triangles like ‚ñ≥BOM and ‚ñ≥DOF, you can express their sides in terms of common factors. If you denote the lengths as *a* and *b*, then you can showcase that the relationship between the triangle sides leads to a direct calculation of ratios such as n:m through cross-multiplication, hence deriving segment ratios for different triangles based on similar principles.