Geometry Chapter 7. Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem

Transcription

1 Geometry Chapter 7 Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem

2 Name: Geometry Assignments Chapter 7 Date Due Similar Polygons Section Topics Assignment 7.1 & 7.2 Ratio Proportion Means Extreme Written Exercises Pg. 244 # odd AND Pg. 247 #4, 6, 8, 20, 22, 24, 26, 28, 33 & & 7.4 Similarity Similar Polygons Scale Factor AA Similarity 7.5 SAS Similarity Postulate SSS Similarity Postulate 7.6 Similarity Proofs Similarity Proofs (cont) & REVIEW Pg. 250 (bottom)-251 # 2-14 even, 15-22, AND Pg. 257 #2-20 even Pg. 266 #2-10 even, 14 AND Pg. 258 # 21-25, 27 Proportional Lengths Triangle Proportionality Theorem Pg #2-11, 20, 22, 23, 25 Triangle Angle-Bisector Theorem Similarity Proofs Worksheet Similarity Proofs Worksheet Chapter Test Remember, if you have any questions or are having difficulty, please come in for extra help. 1

3 x 5 5x 12 2x 3 92 A B 12 D C 2

4 a c b d a c b d ad bc a b c d b d a c a b c d b d 3

5 5x x x 14 x x x4 x5 2 5x 2x1 8 4 AG AH GB HC A G H B C 4

6 Ratio and Proportion WS Geometry 7-1 and 7-2 Name Date Block Solve each proportion for x x x x 4. 6 x x x4 x x 3 x x x x x x x3 x x 7 2 x

7 Complete each proportion. [Use properties!] 13. Given: w 9 x Given: 6 : : 7 a) w 9 a) yh b) x w b) 6 h c) 9x c) 6 y y d) w x x y d) Three numbers aren t known, but the ratio of the numbers is 1 : 3 : 8. Is it possible that the numbers are: a) 1, 3, and 8? b) 3, 9, and 21? c) 10, 30, and 80? d) y, 3y, and 8y? e) x, 3y, and 8z? More practice: p. 246 Class Exercises (1-12) Challenge: 13, 14 6

8 GEOMETRY Notes 7.3 DATE: LOOKING BACK Given the statement, If today is Monday, then I have school. Write a. the contrapositive b. the inverse c. the converse Name the quadrilateral: I have four congruent sides and no right angles. I have two congruent segments and the other pair of sides are parallel. Simplify 18a 9x 6y x 5y 2 2 x y NOTES 7.3 SIMILAR POLYGONS Two polygons are similar if their vertices can be paired so that: a. b. D P C Q E T A Notation B S R If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the. For the example on the page before, the is = The ratio of the perimeters of two similar polygons is. 7

9 Similar Polygons WS1 [notes] Geometry section 7-3 Name Date Block Determine whether the polygons are similar. Explain your reasoning. In each question, the given polygons are similar. Find the value of x

10 In each exercise below, determine whether the polygons are similar. Explain your reasoning. If the polygons are similar, write a similarity statement Complete each of the following An architect is making plans for a rectangular office building that is 840 feet long and 252 feet wide. A blueprint of the floor plan for the first floor is 15 inches long. How wide is the blueprint? 9

11 p.250 CE (1-9) Are the quadrilaterals similar? If they aren't, tell why not. 1. ABCD and EFGH 2. ABCD and JKLM 3. ABCD and NOPQ 4. JKLM and NOPQ 5. If the corresponding angles of two polygons are congruent, must the polygons be similar? 6. If the corresponding sides of two polygons are in proportion, must the polygons be similar? 7. Two polygons are similar. Do they have to be congruent? 8. Two polygons are congruent. Do they have to be similar? 9. Are all regular pentagons similar? Why? 10. JUDY ~ J'U'D'Y'. Complete. a) m<y' = and m<d = b) The scale factor of JUDY to J'U'D'Y' is c) Find DU, Y'J', and J'U'. d) The ratio of perimeters is e) Explain why it is not true that DUJY ~ Y'J'U'D'. 10

12 7-4 Notes: Similarity Proof Similar Polygons: Corresponding angles of similar polygons are congruent AND corresponding sides are proportional. Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Called Example: Given : B A I G B I C We can conclude: H A G 11

13 Algebra first. 12

14 Similar Triangles WS2 Name Geometry 7-3 & 7-6 Date Block Part 1: Use properties of similar triangles to set up equations and solve as needed. Show your work!

15 Part 2: Proving the Triangle Proportionality Theorem. Fill in the following proof Statements Reasons. 1) DE BC given 2) ADE ABC and AED ACB 3) ABC ADE AA similarity 4) AB DA CA EA 5) AD + DB = AB; AE + EC = AC 6) AD DB AE EC DA EA 7) BD DA CE EA If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. 14

16 Part 3: Use similar triangles or the new theorem from part 2 (triangle proportionality theorem) to solve for the value of x in each question. 15

17 2) Given: <QVR <S Prove: (QR)(ST) = (QT)(VR) 16

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19 p.271 CE (3-5). Write a proportion for each question. Solve question 4 & 5. p.272 WE(6-8). Find the value of x. 18

20 7-4 & 7-5 Triangle Similarity Proofs AA Similarity Theorem: SSS Similarity Theorem: SAS Similarity Theorem: Similarity Proofs WS1 Name Geometry Date Block 1. Given: AB DC Prove: AE DE CE BE A E B D C 2. Given: ABCD is a parallelogram Prove: AF EF DF BF B C A F D E 3. Given: ABC; D midpoint of AC; E midpoint of BC Prove: CD DE CA AB [hint : segment DE must be a midsegment ] D C E A B 19

21 ABC with AB AC; PD AB; 4. Given: AE BC Prove: EC AC DB PB D A B E C 5. Find the value of x. given: BC CA; CD BA and lengths in diagram [use similar triangles] C x P 6. If the ratio of the lengths of the segments formed on a hypotenuse of a right triangle from the intersection of the altitude is 1:9; and the length of the altitude to the hypotenuse is 6, find the lengths of the two segments formed on the hypotenuse. [use similar triangles] B 6 D 24 A 7. Find x. given: BD AC; AB BC; AC 50 and lengths in diagram [use similar triangles] A 30 x D B C 8. Find x. given: the two horizontal segments are parallel; not to scale! x 9. Find x. given: the three vertical segments are parallel x x x

22 Similarity Proof WS2 Geometry Name Date Block 21

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25 Similarity Proofs WS3 Geometry Name Date Block VZ WZ 1. Given: A C 2. Given: XZ YZ Prove: ABX CDX Prove: 1 2 B D Z X X V W Y A C 3. Given: PST PRQ 4. Given: Prove: PS PT Prove: PR PQ P BA BC; AE BC; BD AC AC AE BA BD B S T E R Q A D C 5. Given: EDA DAC 6. Given: 1 A 7. Given: 1 2 Prove: BED BAC Prove: ABE CDE Prove: MLN JLK B B 1 C L E D E M 1 N A C A D J 2 K 24

26 Chapter 7 Review Geometry Name Date Block 1. Two angles are complementary and are in the ratio of 7:8. Find the value of the smaller angle. 2. Quad ABCD with m<a: m<b: m<c : m<d = 7:2:2:7. Find the measure of all the angles in the quadrilateral and state what special type of quadrilateral ABCD is. 3. A hexagons angles are in the ratio of 4 : 3 : 7 : 3 : 6 : 7, find the measure of the largest angle. 4. Solve the following proportions for all variables. a x6 x3 b. 12 x x 4 5 c. x7 x13 2 x 2 d. 4 x x 9 e. 3 y x x and 3 2 x 5 y Determine whether the given figures are sometimes, always or never similar. a. Two Rectangles b. Two Squares c. Two isosceles trapezoids with congruent base angles d. Two regular dodecagons e. A scalene triangle and an isosceles triangle f. Two rhombi with at least one < congruent g. Two triangles with proportional sides 6. Quad ABCD ~ Quad HIJK a. If, AB = 8, JK = 12, and CD = 9, find HI. b. If m<b = 50, find m<i. c. If the perimeter of Quad HIJK is equal to 36, what is the perimeter of Quad ABCD? 7. If 2500 square feet of grass emits enough oxygen for a family of 4, how much grass is needed to supply oxygen for a family of five? 25

27 8. If a car uses 15 gallons of gas to travel 500 miles, how many miles does it get for one gallon of gas? 9. Prove whether or not the following triangles are similar. Justify your answer Find x Find w, x, y, z. [diagram not to scale!] 12. Use the diagram below to answer each question. Segment AC is an angle bisector. a. If BC = 20, AB = 18, AD = 45 find CD. b. If AB = 12, AD = 15, and BD = 9, find CD. 13. Given: AC AB, BV AB 14. Given: Rectangle ABCD, EFB CGE Prove: AC JB BV CJ Prove ABG ~ DCF 26

28 Given: Iso. Trap. ABFC with AC BF, AD // Prove: AD DF EF ED BF 19. Given: 1 2 Prove: A D 27

29 EXTRA PRACTICE See me for answers: Pg 252 (Self Test #1) #1-6 Pg 274 (Self Test #2) #1-11 Pg (Chpt Rev) #1-24 Pg 279 (Chpt Test) #