Understanding Ratios, Proportions, and Similarity in Triangles
In this chapter, we explore the fundamental concepts of ratios and proportions, essential for understanding similarity in polygons and triangles. A ratio compares two numbers through division, while proportions demonstrate equivalent ratios through equations. We delve into the properties of similar polygons, the significance of scale factors, and critical postulates and theorems related to triangle similarity. Key topics include cross products, proportional parts, and the relationship between corresponding sides and angles in triangles, enriching your understanding of geometry.
Understanding Ratios, Proportions, and Similarity in Triangles
E N D
Presentation Transcript
Chapter 9 Proportions and Similarity
Section 9-1 Using Ratios and Proportions
Ratio • A ratio is a comparison of two numbers by division
Proportion • An equation that shows two equivalent ratios
Cross Products • The cross products in a proportion are equivalent
Means and Extremes In the proportion 20 = 2 30 3 20 and 3 are the extremes and 30 and 2 are the means.
Theorem 9-1 • If a = c, then ad = bc. b d
Section 9-2 Similar Polygons
Similar Polygons • Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.
Scale Drawing • Used to represent something that is too large or too small to be drawn at actual size.
Scale Factor • The ratio of the lengths of two corresponding sides of two similar polygons
Section 9-3 Similar Triangles
Postulate 9-1 • If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar.
Theorem 9-2 • If the measures of the sides of a triangle are proportional to the measures of the corresponding sides of another triangle, then the triangles are similar.
Theorem 9-3 • If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar.
Section 9-4 Proportional Parts and Triangles
Theorem 9-4 • If a line is parallel to one side of a triangle and intersects the other two sides, then the triangle formed is similar to the original triangle.
Theorem 9-5 • If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths.
Section 9-5 Triangles and Parallel Lines
Theorem 9-6 • 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.
Theorem 9-7 • If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side, and its measure equals one-half the measure of the third side.
Section 9-6 Proportional Parts and Parallel Lines
Theorem 9-8 • If three or more parallel lines intersect two transversals, they divide the transversals proportionally.
Theorem 9-9 • If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
Section 9-7 Perimeters and Similarity
Theorem 9-10 • If two triangles are similar, then the measures of the corresponding perimeters are proportional to the measures of the corresponding sides.
