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Geometry Unit 7 Proportions and Similarity UnitVI7 Unit Ratios and Proportions Ratio Proportion a to b a is to b as c is to d a:b a : b :: c : d a b a c b d UnitVI7 Unit Properties of Proportions Cross-Product Property If a c b d , then ad bc Reciprocal Property If a c b d , then b d a c Congruent Proportions a c b d a b c d d c b a b d a c a b c d b d ab cd b d UnitVI7 Unit Similar Polygons • A D, B E, C F • AC AD BC DF DE EF F C A B D E UnitVI7 Unit Similar Triangles • Angle-Angle (AA) Similarity Postulate – if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar E B A C D F UnitVI7 Unit Similar Triangles • Side-Side-Side (SSS) – if the corresponding sides of two triangles are proportional, then the triangles are similar E B A C D F UnitVI7 Unit Similar Triangles • Side-Angle-Side (SAS) – if an angle of one triangle is congruent to an angle of a second triangle, and the sides including these angles are proportional, then the triangles are similar E B A C D F UnitVI7 Unit Transversal Proportionality • Transversal Proportionality Theorem – Parallel lines divide transversals proportionally. l m n a b a AC b AC c d c BD d BD A a B l c m b C d D n UnitVI7 Unit Similar Triangles • Given ABC DEF – A D, B E, C F AC AD BC – DF DE EF F C A B D E UnitVI7 Unit Fractals • A geometric figure in which a pattern is repeated so that certain parts of the figure are similar to each other. UnitVI7 Unit Special Segments of Similar Triangles • Altitudes – the lengths of corresponding altitudes are proportional to the lengths of the corresponding sides ABC FGH , then A AD AB FJ FG F C D B H J G UnitVI7 Unit Special Segments of Similar Triangles • Angle bisectors – the lengths of corresponding angle bisectors are proportional to the lengths of the corresponding sides KLM QRS , then R LP LM RT RS L Q T S K P M UnitVI7 Unit Special Segments of Similar Triangles • Medians – the lengths of corresponding medians are proportional to the lengths of the corresponding sides ABC WXY , then C CD AB YZ WX Y A D B W Z X UnitVI7 Unit Triangle Angle Bisector Theorem • an angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides R ST SR QT QR Q T S UnitVI7 Unit Ratios of Figures • Ratio of Perimeters: equal to ratio of sides – a:b • Ratio of Areas: equal to ratio of sides squared – a2 : b2 • Ratio of Volumes: equal to ratio of sides cubed – a3 : b3 UnitVI7 Unit