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Ch. 8 Notes 1 Name _________________________________________________________ Hour __________ Warm-up: Simplify each fraction. 1. 24 36 2. 49 56 Students will be able to solve proportions and ratios. Ratio 8.1 & 8.2 Ratio and Proportion ________________________________________ ________________________________________ Proportions An equation that sets two ratios equal a c b d Extremes The numbers a and d Mean The numbers b and c Examples Simplify the ratios. 1. 12cm 4m 2. 6 ft 18in Ch. 8 Notes 2 Name _________________________________________________________ Hour __________ Example The perimeter of rectangle ABCD is 60 centimeters. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. D C w A l B Cross Product Property The product of the extremes equals the product of the means. Reciprocal Property If two ratios are equal, then their reciprocals are also equal. If a c b d , then ad = bc. a c If b d , then More Properties of Proportions Geometric Mean b d a c a c a b c d a c a b c d b d If b d , then If b d , then If the sides of a triangle are in the following a x proportion x b , then the geometric mean is x a b Ch. 8 Notes 3 Name _________________________________________________________ Hour __________ Examples Solve the proportions. 1. 4 5 x 7 The measure of the angles in JKL are in the extended ratio of 1:2:3. Find the measures of the K angles. J L The ratios of the side lengths of DEF to the corresponding side lengths of ABC are 2:1. Find C the unknown lengths. 3 in A B F D Homework: 8.1 worksheet 8 in E Ch. 8 Notes 4 Name _________________________________________________________ Hour __________ Warm-up: Solve each proportion. 1. 12 40 x 20 2. 7 3 9 x Students will be able to solve equations using proportions. Examples 8.2 Problem Solving with Proportions Tell whether the statement is true. a. If p r p 3 , then 6 10 r 5 b. If a c a 3 c 3 , then 3 4 3 4 In the diagram AB AC BD CE . Find the length of BD . A 16 B x D Homework: 8.2 wkst. 30 C 10 E Ch. 8 Notes 5 Name _________________________________________________________ Hour __________ Warm-up: Find the geometric mean of the two numbers. 1. 8 and 18 2. 10 and 15 Students will be able to solve for missing sides of polygons. Similar Polygons 8.3 Similar Polygons ________________________________________ ________________________________________ Theorem 8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their P corresponding side lengths. K Q If KLMN ~ PQRS, then KL LM MN NK KL LM MN NK PQ QR RS SP PQ QR RS SP L S N Examples R M Pentagons JKLMN and STUVW are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of V proportionality. M W N L J K U S T Ch. 8 Notes 6 Name _________________________________________________________ Hour __________ Decide whether the figures are similar. If they are similar, write a similarity statement. Q 4 R X 6 Y 6 10 9 S P 15 8 Z 12 W Quadrilateral JKLM is similar to quadrilateral z Q PQRS. J 10 K 6 Find the value of z. 15 M R S L Parallelogram ABCD is similar to parallelogram GBEF. Find the value of y. B E C 12 G 15 F A P y 24 D Ch. 8 Notes 7 Name _________________________________________________________ Hour __________ UVW is similar to YXW. Find the value ofU a. V a 5 W 6.4 12 X Y Homework: 8.3 wkst Warm-up: Quiz 8.1-8.3 List all the pairs of congruent angles and write the statement of proportionality for the figures. 1. ATL~ NYC Warm-up: Determine whether the following polygons are similar. If so, write a statement of similarity. I 5 L A D O 15 A 1. 2. 4 4 8 6 B 7 C 3 E 4 3.5 F 8 G 15 8 T B 5 Y Students will be able to use the properties of similar triangles to find unknown parts of triangles. Ch. 8 Notes 8 Name _________________________________________________________ Hour __________ 8.4 Similar Triangles Angle-Angle Postulate 25 If two angles of one triangle are congruent to Similarity two angles of another triangle, then the two Postulate triangles are similar. (AA) If JKL XYZ and KJL K YXZ , Y Then JKL ~ XYZ L Z X J Side-SideSide Similarity Postulate (SSS) Theorem 8.2 If the corresponding sides of two triangles are proportional, then the triangles are similar.P AB BC If PQ QR CA , RP A Then ABC ~ PQR Q C B Side-AngleSide Similarity Postulate (SAS) R Theorem 8.3 If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. If X M and ZX XY PM MN Then XYZ ~ MNP , X M N Y Z P Ch. 8 Notes 9 Name _________________________________________________________ Hour __________ In the diagram, BTW ~ ETC . a. Write the statement of proportionality. Examples: T b. E Find mTEC 34o 3 20 C 79o B c. 12 W Find ET and BE. Find the length of the altitude QS . N 12 M 12 6 Q R Homework: 8.4 wkst Warm-up: Find the value of the missing sides. O 24 x G A 12 30 T I 3 B 9 w L y Y 8 S 8 T P Ch. 8 Notes 10 Name _________________________________________________________ Hour __________ Students will be able to prove triangles similar. Examples 8.5 Proving Triangles Which of the following three triangles are similar? A 12 C E G 4 6 9 B 14 6 F J 6 8 10 D H Use the given lengths to prove RST ~ PSQ. Given: SP = 4, PR = 12, SQ = 5, QT = 15 S Prove: RST ~ PSQ 4 P 12 R 5 Q 15 T Ch. 8 Notes 11 Name _________________________________________________________ Hour __________ Indirect Measurement: To measure the width of a river, you use a surveying technique, as shown in the diagram. Use the given lengths (measured in feet) to find RQ. P 63 Q R 12 T 9 S Homework: 8.5 wkst Warm-up: Determine if the following triangles are congruent. If so, write the statement of similarity and give the reason. A D 1. 4 3 B 5 1.5 2 E F C 2.5 Students will be able to use the properties of similar triangles to find unknown parts of triangles. 8.6 Proportions and Similar Triangles Triangle Theorem 8.4 Proportional If a line parallel to one side of a triangle ity Theorem intersects the other two sides, then it divides the Q T two sides proportionally. If TU QS , then RT RU TQ US R S U Ch. 8 Notes 12 Name _________________________________________________________ Hour __________ Converse of Theorem 8.5 If a line divides two sides of a triangle Triangle Proportional proportionally, then it is parallel to the third side. RT RU Q ity Theorem T If TQ US , then TU QS . R S Theorem 8.6 U If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r s and s t , and l and m intersect r, s, and t, Then UW VX WY XZ r U s W X V t Y Z l m Theorem 8.7 If a ray bisects an angle of a triangle it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. If CD bisects ACB , then AD CA A DB CB C D B Ch. 8 Notes 13 Name _________________________________________________________ Hour __________ In the diagram, AB ED , BD = 8, DC = 4, and AE = 12. What is the length of EC ? C 4 D 8 E 12 B A Given the diagram, determine whether MN GH M G 21 56 L N 16 H 48 In the diagram, 1 2 3 , and PQ = 9, QR = 15, and ST = 11. What is the length of TU ? P 9 Q 15 R 3 S 1 2 11 T U Ch. 8 Notes 14 Name _________________________________________________________ Hour __________ In the diagram, CAD ~ DAB . Use the given side lengths to find the length of DC . B 9 D A In the diagram variables. KL MN . 14 C 15 Find the values of the J 9 K L 37.5 7.5 x 13.5 M Homework: 8.6 wkst y N Ch. 8 Notes 15 Name _________________________________________________________ Hour __________ Warm-up: Quiz 8.4-8.6 Are the following triangles similar? If so write the similarity statement and give the reason why. A D 1. o 98 50 o 32 B o C o E 50 F Warm-up: Find the value of the variable. J 1. 9 32 K L 12 x 15 M y N Warm-up: Find the value of the variable. B 1. 10 D 8 A 21 C Warm-up: Ch. 8 Test