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= = =__Chapter 8 Similar Shapes__= = =

__8.1 Dilations and Scale Factors__
Objectives-
 * Construct a dilation of a segment and a point by using a scale factor
 * Construct a dilation of a closed plane figure

__8.2 Similar Polygons__
Objecives-
 * Define similar polygons
 * Use Properties of scale factors to solve problems involving similar polygons

Similar Figures
Two figures are similar if and only if one is congruent to the image of the other by a dilation.

Polygon Similarity Postulate
Two polygons are similar if and only if there is a way of setting up a correspondence between their sides and angles such that the following conditions met:
 * Each pair of corresponding angles is congruent
 * Each pair of corresponding sides is proportional

Properties of Proportions
Let //a, b, c,// and //d// be any real numbers

Cross-Multiply Property
If //a/b=////c/d// and //b// and //d// 0, then ad=bc.

Reciprocal Property
If a/b=c/d and //a, b, c//, and //d// 0, then b/a=d/c.

Exchange Property
If a/b=c/d and //a, b, c//, and //d// 0, then a/c=b/d.

"Add-One" Property
If //a/b//=//c/d// and //a, b, c//, and //d// 0, a+b/b=c+d/d. [|link to picture]

__8.3 Triangle Similarity__
Objective-
 * Develop the AA Triangle Similarity Postulate and the SSS and SAS Triangle Similarity Theorems.



AA (Angle-Angle) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

SSS (Side-Side-Side) Similarity Theorem
If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

SAS Similarity Theorem
If two sides of the triangles are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

__8.4 The Side Splitting Theorem__
Objectives-
 * Develop and prove the side-splitting Theorem
 * Use the Side-Splitting Theorem to solve problems

Side Splitting Theorem
A line parallel to one side of the triangle divides the other two sides proportionally.

Two Transversal Proportionality Corollary
Three or more parallel lines divide two intersecting transversals proportionally.

__8.5 Indirect Measurement and Additional Similarity Theorems__
Objective-
 * Use triangle similarity to measure distances indirectly.
 * Develop and use similarity theorems for altitudes and medians of triangles.

Proportional Medians Theorem
If two triangles are similar, then their corresponding medians have the same ratio as their corresponding sides.

Proportional Angle Bisectors Theorem
If two triangles are similar, then their corresponding angle bisectors have the same ratio as the corresponding sides

__8.6 Area and Volume Ratios__
Objectives-
 * Develop and use ratios for areas of similar figures.
 * Develop and use ratios for voums of similar sides
 * Explore relationships between cross-sectional area, weight, and height.