Similarity, Ratio and proportion

In arithmetic we learn that one method of comparing two quantities in respect of their magnitude is to express them in the form of a fraction the numerator and denominator of which state the sizes of the quantities measured in suitable and the same units. This form of comparison is called a ratio.
Hence we speak of the ratio of two straight lines, we mean the ratio of the numbers which express the measures of their lengths in terms of the same unit. Similarly, by the ratio of the areas of two triangles we means the ratio of the numbers which express these areas in the same square units.
Similar Triangles:- The two triangles, however, have the same shape one might be considered an enlargement of the other. Such triangle are called similar triangle and to indicate this, we write triangle ABC similar to triangle DEF. This notation indicates a special correspondence between the vertices of the two triangles, just as the notation used for congruence does.
- Two triangles are similar iff there is a correspondence berween their vertices such that the corresponding sides of the triangles are proportional and the corresponding angles are equal.
- If correspondence of triangles is such that two angle of one triangle are congruent to their corresponding angles, then the third pair of corresponding angles are congruent and the correspondence is a similarity.
- If, in a correspondence of triangles, two sides are proportional to their corresponding sides and he corresponding angle between these sides are congruent, then the correspondence is a similarity.
- If in a correspondence of triangles, the three sides of one triangle are proportional to the corresponding sides in the other triangle, then the correspondence is a similarity.

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