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Geometry Chapter 7 Study Guide (7.1-7.6) - Similarity Vocabulary ratio - compares two numbers by division proportion - an equation stating two ratios are equal similar polygons - two polygons with congruent corresponding angles and proportional side lengths similarity ratio - the ratio of the corresponding sides of two similar polygons dilation - a transformation that changes the size of a figure but not its shape scale factor - describes how much the figure is enlarged or reduced Theorems, Postulates, and Properties Angle-Angle Similarity Postulate (AA~) - If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Side-Side-Side Similarity Theorem (SSS~) - If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. Side-Angle-Side Similarity Theorem (SAS~) - If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. • Triangle Proportionality Theorem - If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. • Triangle Angle Bisector Theorem - An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. • Two Transversal Proportionality Corollary - If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. a • Proportional Perimeters and Areas Theorem - If the similarity ratio of two similar figures is ! b , then the ratio of their 2 perimeters is ! ba and the ratio of their areas is ! ba2 or ! • Properties of Similarity • Reflexive Property of Similarity • Transitive Property of Similarity ! ΔABC ( ba )2 ∼ ΔABC If ! ΔABC ∼ ΔDEF and ! ΔDEF ∼ ΔXYZ , then ! ΔABC ∼ ΔXYZ . Geometry Chapter 7 Practice Problems 1) Two rectangles are similar. They both have a width to length ratio of 2:3. The ratio of the lengths between the rectangles is 3:1. The larger rectangle has a perimeter of 90 miles. Find the area of each rectangle. Name: ______________________________________ Date: ____________________ Period: ____________ 2) Explain why the triangles are similar and find the length of DE. ! Reason: Area of smaller = ___________ Area of larger = ____________ DE = ___________ 3) Find KN and LM. 4) Find the lengths BE and DE. ! ! KN = _______ BE = ________ LM = _______ DE = ________ 5) In ! ΔQRS , the bisector of ! ∠R divides ! QS into segments with lengths 2.1 and 2.8. If RQ = 3, what is the 6) Given that ! DEFG area of WXYZ. ∼ WXYZ , find the perimeter and length of ! RS ? Draw a diagram and solve. ! RS = _________ Perimeter = __________ 7) A free-fall ride at an amusement park casts a shadow Area = ___________ 8) Given that ! ΔPQR ∼ ΔPST , find the scale factor and the coordinates of S. ! 43 2 feet long. At the same time, a 6-foot-tall person 3 standing in line casts a shadow 2 feet long. What is the height of the ride? ! Scale factor = ______ Height of ride = __________ S = ___________