Topic 7: Similarity 7-1: Properties of Proportions 7-2: Applications of Proportions 7-3: Similar Polygons 7-4: Similar Triangles 7-5: Proportional Parts 7-6: Parts of Similar Triangles Home Next

7-1: Properties of Proportions Definition: A ratio is a comparison between two numbers that can be expressed three different ways If a and b are Real Numbers and b  0, a to b, a:b, and all mean the same thing Definition: Two equal ratios are called a proportion and are examples of proportions Topic 7 Next

7-1: Properties of Proportions Equality of Cross Products: For any numbers a and c and any nonzero numbers b and d, if and only if ad = bc. In the proportion notice that Example: a c d b = a and d are called the extremes b and c are called the means Given: In a proportion the product of the extremes equals the product of the means. Back Section 7-2

7-2: Applications of Proportions Definition: A rate is a ratio that compares two different types of units If you travel 240 miles in 4 hours your average rate is 60 miles per hour Example: Expressed as a Proportion: Notice In the Proportion Topic 7 Section 7-3

7-3: Similar Polygons Definition: Topic 7 Section 7-4 Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. Usually similar objects have the same shape but are different sizes If ABCD is similar to WXYZ then Sides are Proportional Scale Factor The ratio of corresponding sides is called the scale factor. Topic 7 Section 7-4

7-4:Similar Triangles Angle-Angle (AA) Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.   If A  A, and AED  ACB, then *ACB  AED * The symbol  is used to indicate similarity SSS Similarity Theorem: If the measures of corresponding sides of two triangles are proportional, then the triangles are similar If then ACB  AED Topic 7 Next

7-4: Similar Triangles SAS Similarity Theorem: If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. If and B  Y, then ABC  XYZ Back Next

7-4: Similar Triangles Theorem: Similarity of Triangles is Reflexive, Symmetric and Transitive. If ABC  DEF, and DEF  GHI, then ABC  GHI If ABC  DEF, then DEF  ABC ABC  ABC Transitive Property Symmetric Property Reflexive Property Back Section 7-5

7-5: Proportional Parts Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths. If DE ll CB, then Theorem: If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. If , then DE ll CB Topic 7 Next

7-5: Proportional Parts Theorem: A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. If W is the midpoint of VX and Y is the midpoint of XZ, then WY ll VZ and Back Next

7-5: Proportional Parts Corollary: If p ll q ll r then, AND Back Next If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally If p ll q ll r then, AND Back Next

7-5: Proportional Parts Corollary: If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal If p ll q ll r and AB  BC, then XY  YZ. Back Section 7-6

7-6: Parts of Similar Triangles Proportional Perimeter Theorem: If two triangles are similar, then their perimeters are proportional to the measures of corresponding sides. Let P1 = a + b + c, the perimeter of ABC Let P2 = d + e + f, the perimeter of DEF If ABC  DEF, then Topic 7 Next

7-6: Parts of Similar Triangles Theorem: If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides If ABC  DEF and CP and FQ are corresponding altitudes, then Back Next

7-6: Parts of Similar Triangles Theorem: If two triangles are similar, then the measures of the corresponding angle bisectors are proportional to the measures of the corresponding sides If PQR  XYZ and QG and YH are corresponding angle bisectors, then Back Next

7-6: Parts of Similar Triangles Theorem: If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides If PQR  XYZ and RF and ZG are corresponding medians, then Back Next

7-6: Parts of Similar Triangles Theorem: An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. Given ABC with 1   2 2 1 Given ABC with CD the angle bisector of ACB, Back Right Triangles & Trigonometry