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Geometry Chapter 7 Ratios & Proportions Properties of Proportions Similar Polygons Similarity Proofs Triangle Angle Bisector Theorem Name: _________________________________________________________________ Geometry Assignments – Chapter 7 Similar Polygons Date Due Section Topics Assignment Written Exercises 7.1 & 7.2 7.3 & 7.4 7.5 Similarity Similar Polygons Scale Factor AA Similarity SAS Similarity Postulate SSS Similarity Postulate Proportional Lengths Triangle Proportionality Theorem Triangle Angle-Bisector Theorem Similarity Proofs Similarity Proofs 7.6 Similarity Proofs Similarity Proofs (cont) & REVIEW Ratio Proportion Means Extreme Pg. 244 # 25-31 odd AND Pg. 247 #4, 6, 8, 20, 22, 24, 26, 28, 33 & 37 Pg. 250 (bottom)-251 # 2-14 even, 15-22, 24-27 AND Pg. 257 #2-20 even Pg. 266 #2-10 even, 14 AND Pg. 258 # 21-25, 27 Pg. 272-273 #2-11, 20, 22, 23, 25 Worksheet Worksheet Chapter Test Remember, if you have any questions or are having difficulty, please come in for extra help. 1 x 5 5x 12 2 x 3 92 15 A 105 B 145 12 C D 2 a c b d 2 8 3 12 a c b d ad bc a b c d b d a c ab cd b d 3 4 3 95 6 x 5 x 35 24 9 5.4 6 x 7.2 14 x x 56 3 7 x 4 x 5 2 5x 2 x 1 8 4 A AG AH GB HC G B H C 4 Ratio and Proportion WS Geometry – 7-1 and 7-2 Name __________________________ Date _______________ Block ______ Solve each proportion for x. 1. 6 x 27 24 9 2. 4.8 6 x 8.4 3. 4. 6 x x 150 5. 3 7 x4 x4 6. 7. 10. x 22 18 12 8 x x 50 8. 11. 22 2 x 18 3 5 x 3 x 1 2 7 5 10 8 x 5 2 x 3x 1 8 4 5 8 x 9. 1 4 16 12. 3x 7 2 x 1 15 21 5 ________________________________________________________________ Complete each proportion. [Use properties!] w 9 13. Given: 14. Given: 6 : y h : 7 x 17 w a) a) y h 9 x 6 b) b) w h 6 y c) 9x c) y w x y d) d) x 6 __________________________________________________________________ 15. 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 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 36 5x 5 y x2 y 2 9x 6 y 3 NOTES – 7.3 – SIMILAR POLYGONS Two polygons are similar if their vertices can be paired so that: a. b. P D C E Q 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 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. 5. 6. 7. 8. 8 In each exercise below, determine whether the polygons are similar. Explain your reasoning. If the polygons are similar, write a similarity statement. 9. 10. 11. 12. Complete each of the following. 13. 14. 15. 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 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 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 I B I A G C We can conclude: H A G 11 Algebra first…. 12 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! 1. 2. 3. 4. 13 5. 6. 7. ________________________________________________________________________ Part 2: Proving the Triangle Proportionality Theorem. Fill in the following proof… Statements 1) DE BC 2) ADE 3) ABC 4) Reasons given ABC and AED ADE . ACB AA similarity AB CA DA EA 5) AD + DB = AB; AE + EC = AC 6) AD DB AE EC DA EA 7) BD CE DA EA If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. 14 ________________________________________________________________________ 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 2) Given: <QVR ≅ <S Prove: (QR)(ST) = (QT)(VR) 16 17 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 7-4 & 7-5 Triangle Similarity Proofs AA Similarity Theorem: SSS Similarity Theorem: SAS Similarity Theorem: _____________________________________________________________________________ Similarity Proofs WS1 Name _____________________________ Geometry Date _________ Block ___________ B A 1. Given: AB DC Prove: AE DE CE BE E D 2. Given: ABCD is a parallelogram Prove: AF EF DF BF C B C A D F E 3. Given: ABC; D midpoint of AC; E midpoint of BC CD DE Prove: CA AB [hint : segment DE must be a midsegment…] A C D E B 19 4. Given: ABC with AB AC; PD AB; A AE BC EC AC Prove: DB PB D B C E ________________________________________________________________________ C 5. Find the value of x. given: BC CA; CD BA and lengths in diagram x [use similar triangles] B 6 A 24 D 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] 7. Find x. given: BD AC; AB BC; AC 50 and lengths in diagram [use similar triangles] A x D 30 B 8. Find x. given: the two horizontal segments are parallel; not to scale! C 6 10 x 15 9. Find x. given: the three vertical segments are parallel x+1 2 x x+6 20 Similarity Proof WS2 Geometry Name _________________________ Date _____________ Block ______ ______________________________________________________________________ 21 __________________________________________________________________ 22 ______________________________________________________________________ 23 Similarity Proofs WS3 Geometry Name ____________________________ Date ________________ Block ______ 1. Given: A C 2. Given: Prove: ABX VZ WZ XZ YZ Prove: 1 2 CDX B D Z W V X X Y C A BA BC ; 3. Given: PST PRQ 4. Given: AE BC ; PS PT Prove: PR PQ BD AC AC AE Prove: BA BD B P S E T R Q A 5. Given: EDA DAC Prove: BED BAC 6. Given: 1 A Prove: ABE CDE D 7. Given: 1 2 Prove: MLN JLK B L 1 B E A C D M E C C A J 1 N 2 K D 24 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. 9 5 3x 6 x 3 b. 12 x x4 5 d. 4 x x 9 e. y x 3 x x5 y and 3 2 4 5 c. x 7 x 13 2 x2 5. 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 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. 15 5 4 10. Find x. 25 20 3 11. 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 Prove: AC JB BV CJ 14. Given: Rectangle ABCD , EFB CGE Prove ABG ~ DCF 26 15. 16. 17. 18. Given: Iso. Trap. ABFC with AC BF , AD // BF Prove: AD DF EF ED 19. Given: Prove: 1 2 A D 27 EXTRA PRACTICE – See me for answers: Pg 252 (Self Test #1) #1-6 Pg 274 (Self Test #2) #1-11 Pg 277-278 (Chpt Rev) #1-24 Pg 279 (Chpt Test) #1-15 28