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CHAPTER 7 SIMILAR POLYGONS SECTION 7-1 Ratios and Proportions RATIO – a comparison of two numbers, a and b, represented in one of the following ways: a:b a or a to b b EQUIVALENT RATIOS – two ratios that can both be named by the same fraction. 4:8 and 7 :14 PROPORTION – is an equation that states that two ratios are equivalent. a:b=c:d a=c b d EXTREMES – the first and last terms a : b= c : d a and d are extremes MEANS – the second and third terms a:b=c:d b and c are means CROSS PRODUCTS – the product of the extremes equals the product of the means. ad= bc SECTION 7-2 Properties of Proportions TERMS – the four numbers a, b, c, and d that are related in the proportion. Properties of Proportions a/b = c/d is equivalent to: a) ad = bc b) a/c = b/d c) b/a = d/c d) (a + b)/b = (c + d)/d 2. If a/b = c/d = e/f = …, then (a+c+e+…)/(b+d+f+…) = a/b = … 1. SECTION 7-3 Similar Polygons SCALE DRAWING – is a representation of a real object. SCALE – is the ratio of the size of the drawing to the actual size. SIMILAR – figures that have the same shape CORRESPONDING ANGLES – angles in the same position in congruent or similar polygons. CORRESPONDING SIDES – sides in the same position in congruent or similar polygons. SIMILAR POLYGONS – figures having all corresponding angles congruent and the measures of all corresponding sides are in the same proportion. The symbol for similarity is Scale Factor - The ratio of the lengths of two corresponding sides SECTION 7-4 A Postulate for Similar Triangles AA Similarity If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. SECTION 7-5 Theorems for Similar Triangles SAS Similarity If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. SSS Similarity If the sides of two triangles are in proportion, then the triangles are similar. SECTION 7-6 Proportional Lengths Theorem 7-3 If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. Corollary If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem 7-4 If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. END