CHAPTER 7 SIMILAR POLYGONS SECTION 7 1 Ratios

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 n 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 n 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 n If the sides of two triangles are in proportion, then the triangles are similar.

SECTION 7 -6 Proportional Lengths

Theorem 7 -3 n If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Corollary n If three parallel lines intersect two transversals, then they divide the transversals proportionally.

Theorem 7 -4 n If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.

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