B. Examples: 1. At NVHS, there are 104 teachers and 2204 students. What is the approximate teacher to student ratio?

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1 Name Date Period Notes Formal Geometry Chapter 7 Similar Polygons 7.1 Ratios and Proportions A. Definitions: 1. Ratio: 2. Proportion: 3. Cross Products Property: 4. Equivalent Proportions: B. Examples: 1. At NVHS, there are 104 teachers and 2204 students. What is the approximate teacher to student ratio? 2. Kjk 3. Solve each proportion: 4. Fg 1

2 C. Guided Practice: Solve each proportion. (Activity) 7.2 Similar Polygons A. Definitions: 1. Similar Polygons: 2. Similarity ratios: 3. Scale Factor: the ratio of the lengths of the corresponding sides of two similar polygons. The order of comparison matters. 4. Theorem 7.1: Perimeters of Similar Polygons: B. Examples: 1. fg 2

3 2. Fg 3. fg 4. Df C. Guided Practice: 1. Df 2. Df a. x b. y 3

4 7.3 Similar Triangles A. Warm-up: 1. What do you know about the angles of similar triangles? 2. If you know that two angles of one triangle are congruent to two angles of another triangle, are the triangles congruent? Are the triangles similar? Explain your reasoning. B. Definitions: 1. Postulate 7.1: 2. Theorem 7.2: Side-Side-Side (SSS) Similarity: If the corresponding side lengths of two triangles are, then the triangles are. 3. Theorem 7.3: Side-Angle-Side (SAS) Similarity: If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are. 4. Properties of Similarity between Triangles: C. Examples: 1. 4

5 Find BE and AD. 5. kl 7.4 Parallel Lines and Proportional Parts A. Warm-up: 5

6 B. Definitions: 1. Theorem 7.5: Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and the other two sides, then it divides the sides into segments of lengths. 2. Theorem 7.6: Converse of Triangle Proportionality Theorem: 3. Midsegment of a triangle: 4. Theorem 7.7: Triangle Midsegment Theorem: a midsegment of a triangle is parallel to one side of the triangle, and its length is the length of that side. 5. Corollary 7.1: Proportional Parts of Parallel Lines: If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. 6. Corollary 7.2: Congruent Parts of Parallel Lines: If three or more parallel lines cut off, then they cut off congruent segments on every transversal. C. Examples: 1. kl 2. Kl 6

7 3. 4. df 5. Find x and y. 7.5 Parts of Similar Triangles A. Warm-up: 1. Find DE, DB and the measure of angle FED. B. Definitions: 1. Theorem 7.8: If two triangles are similar, the lengths of corresponding are proportional to the lengths of the corresponding sides. 7

8 2. Theorem 7.9: If two triangles are similar, the lengths of corresponding are proportional to the lengths of the corresponding sides. 3. Theorem 7.10: If two triangles are similar, the lengths of corresponding are proportional to the lengths of the corresponding sides. 4. Theorem 7.11: Triangle Angle Bisector: C. Examples: 1. fg 2. Fgf Additional information: Liliana is standing and holding her arm straight out in front with her thumb up and at eye-level. The length of Liliana s arm is 10 times the distance between her eyes. Closing one eye, she aligns the edge of the thumb with the edge of the clock. (See p497 in text for figure.) 3. Solve for x. 8

9 9.6 Dilations YOU NEED A RULER! A. Warm-up: 1. A replica of the Statue of Liberty in Austin, TX is feet tall. If the scale factor of the actual statue to the replica is 9:1, how tall is the statue in New York Harbor? 2. The Eiffel Tower in Paris, France is 986 feet tall. A replica of the Eiffel Tower was built as a ride in an amusement park. If the scale factor of the actual tower to the replica is 3:1, how tall is the ride? B. Definitions: 1. Dilation: 2. Isometry Dilation: 3. Dilations in the Coordinate Plane: C. Examples: 1. 9

10 The Pythagorean Theorem and Its Converse D. Definitions: Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Example: 10

11 Prove the Pythagorean Theorem using Geometric Means: Given: ABC is a right triangle with legs of length a and b and hypotenuse of length c. Prove: a 2 + b 2 = c 2 Part I. Prove triangles similar: Draw a perpendicular from C to the hypotenuse. Label the point of intersection X. 1) BXC BCA because 2) B B by 3) So, BXC~ BCA by 4) AXC ACB because 5) A A by 6) So, AXC~ ACB by Part II. Write proportions using Geometric Mean: Label AX = d and label XB = e. Use the fact that corresponding sides of similar triangles are proportional to write two proportions. Proportion 1: Proportion 2: Part II. Set up the equations and solve. Start with getting a and b by itself in each proportion. (you will have a 2 and b 2 ). Next add the two equations together. Use factoring and the fact that e + d = c to solve for the Pythagorean theorem. 5. Pythagorean Triple: 6. Converse of the Pythagorean Theorem: 11

12 7. Acute Pythagorean Inequality Theorem: If the square of the length of the side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is a triangle. 8. Obtuse Pythagorean Inequality Theorem: If the square of the length of the longest side of a triangle is than the sum of the squares of the lengths of the other two sides, then the triangle is an triangle. E. Examples: 1. Find the missing measures using the Pythagorean Theorem. 2. Use a Pythagorean triple to find each missing length. Explain your reasoning. A B 3. Df c. 12