Skills Practice Skills Practice for Lesson 4.1

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1 Skills Prctice Skills Prctice for Lesson.1 Nme Dte Interior nd Exterior Angles of Tringle Tringle Sum, Exterior Angle, nd Exterior Angle Inequlity Theorems Vocbulry Write the term tht best completes ech sttement. 1. The Exterior Angle Inequlity Theorem sttes tht the mesure of n exterior ngle of tringle is greter thn the mesure of either of the remote interior ngles of the tringle.. The Tringle Sum Theorem sttes tht the sum of the mesures of the interior ngles of tringle is The Exterior Angle Theorem sttes tht the mesure of n exterior ngle of tringle is equl to the sum of the mesures of the remote interior ngles of the tringle.. The remote interior ngles of tringle re the two ngles tht re non-djcent to the specified exterior ngle. Problem Set Determine the mesure of the missing ngle in ech tringle. 1. B. P Q A C R m B 180 (78 37 ) 65 m R 180 (80 66 ) Crnegie Lerning, Inc. Chpter l Skills Prctice 9

2 3. K. G 35 M 8 L 90 F 3 E m L 180 (8 35 ) 117 m G 180 (90 3 ) W 6. T X V 35 U Y m Y 180 (60 60 ) 60 m U 180 ( ) 35 List the side lengths from shortest to longest for ech digrm. 7. C 8. S B 8 b c 1 A r t T 60 s 5 R 010 Crnegie Lerning, Inc. m C 180 (8 1 ) 111 m S 180 (5 60 ) 66 The shortest side of tringle The shortest side of tringle is opposite the smllest ngle. is opposite the smllest ngle. So, the side lengths from shortest So, the side lengths from shortest to longest re, b, c. to longest re r, t, s. 30 Chpter l Skills Prctice

3 Nme Dte 9. k M 10. Z L 8 m 118 K l Y x y 8 X z m M 180 (118 8 ) 3 m Y 180 (8 ) 5 The shortest side of tringle The shortest side of tringle is opposite the smllest ngle. is opposite the smllest ngle. So, the side lengths from shortest So, the side lengths from shortest to longest re l, m, k. to longest re z, y, x. 11. X b Y d c 7 W 6 e Z B u r t A s D v C m X 180 (67 7 ) 86 m A 180 (60 30 ) 90 m Z 180 (6 79 ) 37 m C 180 (90 50 ) 0 The shortest side of tringle is The shortest side of tringle is opposite the smllest ngle. Side c is opposite the smllest ngle. Side t is the longest side of WXY nd the the longest side of ABD nd the shortest side of WYZ. So, the side shortest side of BCD. So, the side lengths from shortest to longest lengths from shortest to longest re b,, c, d, e. re s, r, t, v, u. 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 31

4 Identify the interior ngles, the exterior ngle, nd the remote interior ngles of ech tringle. 13. W X Y 1. T U Z R S Interior ngles: XYZ, YZX, ZXY Exterior ngle: WXZ Remote interior ngles: XYZ, YZX Interior ngles: RST, RTS, SRT Exterior ngle: STU Remote interior ngles: RST, SRT 15. F 16. B E G H C A D Interior ngles: EFG, EGF, FEG Exterior ngle: FGH Remote interior ngles: EFG, FEG Interior ngles: ABC, ACB, BAC Exterior ngle: BAD Remote interior ngles: ABC, ACB 17. L 18. P 010 Crnegie Lerning, Inc. J K M Interior ngles: JKL, JLK, KJL Exterior ngle: LKM Remote interior ngles: JLK, KJL Q R S Interior ngles: QRS, QSR, RQS Exterior ngle: PQS Remote interior ngles: QRS, QSR 3 Chpter l Skills Prctice

5 Nme Dte Solve for x in ech digrm. 19. J 130 F G x 99 H K 0. R x 13 T U 10 S V m GFH m RTS m GHK m GFH m FGH m RSV m RTS m SRT x 10 8 x 9 x 9 x 1. H x I x K J 81. U 6 90 (x + 8) R T V S m IJK m UTV m IJK m HIJ m IHJ m SVU m UTV m TUV 99 x x x x x x x Crnegie Lerning, Inc. Chpter l Skills Prctice 33

6 3. M 13 J (x + ) K. G 90 F 11 L N D (3x + ) (x + 18) E m KJL m DFE m KLN m KJL m JKL m DFG m DEF m EDF 11 8 (x ) 90 (x 18 ) (3x ) 11 5 x 90 5x 0 60 x 70 5x 30 x 1 x Use the given informtion for ech tringle to write two inequlities tht you would need to prove the Exterior Angle Inequlity Theorem. 5. T 6. Q R P Q R S S Given: Tringle RST with exterior TRQ Prove: m TRQ m S nd m TRQ m T Given: Tringle QRS with exterior PQR Prove: m PQR m R nd m PQR m S 010 Crnegie Lerning, Inc. 3 Chpter l Skills Prctice

7 Nme Dte 7. T U 8. J W V F G H Given: Tringle UVW with exterior TUV Prove: m TUV m V nd m TUV m W Given: Tringle GHJ with exterior FGJ Prove: m FGJ m H nd m FGJ m J 9. K L M 30. A B N C D Given: Tringle LMN with exterior KLN Prove: m KLN m M nd m KLN m N Given: Tringle ABC with exterior BCD Prove: m BCD m A nd m BCD m B 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 35

8 Skills Prctice Skills Prctice for Lesson. Nme Dte Instlling Stellite Dish Simplifying Rdicls, Pythgoren Theorem, nd Its Converse Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves rdicl sign. squre root d. rdicl expression. the symbol b. rdicl sign b. rdicl sign 3. number b such tht b c. rdicnd. squre root. the sides of right tringle tht form the right ngle d. rdicl expression g. legs 5. the expression written under rdicl sign e. simplest form 010 Crnegie Lerning, Inc. c. rdicnd 6. when the rdicnd of rdicl expression f. hypotenuse contins no fctors tht re perfect squres e. simplest form 7. the side opposite the right ngle in right tringle g. legs f. hypotenuse Chpter l Skills Prctice 37

9 Problem Set Clculte the vlue of ech rdicl expression () (3) ; ; (6) (10) 50 Simplify ech expression nd write the result in rdicl form Nme the form of 1 tht you would use to simplify ech frction Crnegie Lerning, Inc. 38 Chpter l Skills Prctice

10 Nme Dte Simplify ech frction Given the re A of squre, clculte the length of one side. 010 Crnegie Lerning, Inc. 3. A 8 cm A A Ech side is 3 centimeters long. Chpter l Skills Prctice 39

11 . A 75 m A A Ech side is 5 3 meters long. 5. Ingrid covers the floor of squre room with 196 lrge tiles. The re of ech tile is 1 squre foot. Wht is the length of one side of the room? A A Ech side of the room is 1 feet long. 6. Devon prepres squre grden with n re of 180 squre feet. How much fencing will Devon need for ech side of the grden? A A Devon will need 6 5 feet of fencing for ech side of the grden. Determine the length of the hypotenuse of ech tringle. Round your nswer to the nerest tenth, if necessry c 8. 6 c 8 c 3 c 6 8 c 9 16 c 36 6 c 5 c 100 c 5 5 c The length of the hypotenuse is The length of the hypotenuse is 5 units. 10 units. 010 Crnegie Lerning, Inc. 0 Chpter Skills Prctice

12 Nme Dte 9. 8 c c 8 15 c 8 8 c c 6 6 c c 18 c 35 c 18 c 35 c c The length of the hypotenuse is pproximtely 11.3 units. The length of the hypotenuse is pproximtely 18.0 units. Determine the length of the unknown leg. Round your nswer to the nerest tenth, if necessry b 5 b b b b 1 81 b 1 81 b The length of the unknown leg is 1 units. The length of the unknown leg is 9 units. 010 Crnegie Lerning, Inc. Chpter Skills Prctice 1

13 b b b b b 7 75 b b The length of the unknown leg is pproximtely 8.7 units. The length of the unknown leg is pproximtely 8.5 units. Use the converse of the Pythgoren Theorem to determine whether ech tringle is right tringle. Explin your nswer Yes. This is right tringle The sum of the squres of the lengths of the two legs is equl to the squre of the length of the hypotenuse, so this is right tringle. 010 Crnegie Lerning, Inc. Chpter Skills Prctice

14 Nme Dte No. This is not right tringle The sum of the squres of the lengths of the two legs is not equl to the squre of the length of the hypotenuse, so this is not right tringle No. This is not right tringle The sum of the squres of the lengths of the two legs is not equl to the squre of the length of the hypotenuse, so this is not right tringle Crnegie Lerning, Inc. 0 Yes. This is right tringle The sum of the squres of the lengths of the two legs is equl to the squre of the length of the hypotenuse, so this is right tringle. Chpter Skills Prctice 3

15 Use the Pythgoren Theorem to clculte ech unknown length. Round your nswer to the nerest tenth, if necessry. 39. Chndr hs ldder tht is 0 feet long. If the top of the ldder reches 16 feet up the side of building, how fr from the building is the bse of the ldder? 16 b 0 56 b 00 b b 1 b 1 b 1 The bse of the ldder is 1 feet from the building. 0. A scffold hs digonl support bem to strengthen it. If the scffold is 1 feet high nd 5 feet wide, how long must the support bem be? 5 1 c 5 1 c 169 c 169 c 13 c The length of the digonl support bem is 13 feet. 1. The length of the hypotenuse of right tringle is 0 centimeters. The legs of the tringle re the sme length. How long is ech leg of the tringle? The length of ech leg of the tringle is pproximtely 8.3 centimeters. 010 Crnegie Lerning, Inc. Chpter Skills Prctice

16 Nme Dte. A crpenter props ldder ginst the wll of building. The bse of the ldder is 10 feet from the wll. The top of the ldder is feet from the ground. How long is the ldder? 10 c c 676 c 676 c 6 c The ldder is 6 feet long. 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 5

17 Skills Prctice Skills Prctice for Lesson.3 Nme Dte Specil Right Tringles Properties of Tringle Vocbulry Define ech term in your own words tringle A tringle is n isosceles right tringle Tringle Theorem The Tringle Theorem sttes tht the length of the hypotenuse in tringle is times the length of leg. Problem Set Determine the length of the hypotenuse of ech tringle. Write your nswer s rdicl in simplest form. 1. in. c. 5 cm c in. c c 5 5 cm 3. The length of the hypotenuse is The length of the hypotenuse is inches. 5 centimeters. 9 ft c 9 ft. 7 km c 9 c 7 The length of the hypotenuse is The length of the hypotenuse is 9 feet. 7 kilometers. c 7 km 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 7

18 Determine the lengths of the legs of ech tringle. Write your nswer s rdicl in simplest form cm 6. 1 mi The length of ech leg is 8 centimeters. The length of ech leg is 6 miles ft 8. 8 m The length of ech leg is 6 feet. The length of ech leg is 8 meters. 010 Crnegie Lerning, Inc. Use the given informtion to nswer ech question. Round your nswer to the nerest tenth, if necessry. 9. Soren is flying kite on the bech. The string forms 5º ngle with the ground. If he hs let out 16 meters of line, how high bove the ground is the kite? The kite is pproximtely 11.3 meters bove the ground. 8 Chpter l Skills Prctice

19 Nme Dte 10. Meen is picking ornges from the tree in her yrd. She rests 1-foot ldder ginst the tree t 5º ngle. How fr is the top of the ldder from the ground? The top of the ldder is pproximtely 8.5 feet from the ground. 11. Emily is building squre bookshelf. She wnts to dd digonl support bem to the bck to strengthen it. The digonl divides the bookshelf into two 5º 5º 90º tringles. If ech side of the bookshelf is feet long, wht must the length of the support bem be? c 5.7 The support bem must be pproximtely 5.7 feet long. 1. Prospect Prk is squre with side lengths of 51 meters. One of the pths through the prk runs digonlly from the northest corner to the southwest corner, nd divides the prk into two 5º 5º 90º tringles. How long is tht pth? c The pth is pproximtely 7.1 meters long. 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 9

20 Determine the re of ech tringle mm A 1 (8 )(8 ) A 6( ) A 6() A 6 The re of the tringle is 6 squre millimeters in A 1 (9 )(9 ) A 81( ) A 81() A 81 The re of the tringle is 81 squre inches. 010 Crnegie Lerning, Inc ft A 1 ( 7 )( 7 ) A 9( ) 8 A 9() 8 A 1.5 The re of the tringle is 1.5 squre feet. 50 Chpter l Skills Prctice

21 Nme Dte m A ( 1 11 )( 11 ) A 11( ) 8 A 11() 8 A 30.5 The re of the tringle is 30.5 squre meters. Use the given informtion to nswer ech question. 17. Eli is mking mosic using tiles shped like 5º 5º 90º tringles. The length of the hypotenuse of ech tile is 13 centimeters. Wht is the re of ech tile? ( A ( 1 13 )( 13 ) ) ( ) 13 A 169( ) 169() 8 8 A The re of ech tile is.5 squre centimeters. 18. Bked pit chips re often in the shpe of 5º 5º 90º tringles. Citlyn finds tht the longest side of pit chip in one bg mesures 3 centimeters. Wht is the re of the pit chip? 3 A ( )( 3 ) 3 A 9( ) 8 3 A 9() 8 A.5 The re of ech pit chip is.5 squre centimeters. 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 51

22 19. Annik is mking kite in the shpe of 5º 5º 90º tringle. The longest side of the kite is 8 inches. Wht is the re of the piece of fbric needed for the kite? A 1 (1 )(1 ) A 196( ) A 196() A 196 The re of the piece of fbric needed for the kite is 196 squre inches. 0. A tent hs mesh door tht is shped like 5º 5º 90º tringle. The longest side of the door is 36 inches. Wht is the re of the mesh door? A 1 (18 )(18 ) A 3( ) A 3() A 3 The re of the mesh door is 3 squre inches. 010 Crnegie Lerning, Inc. 5 Chpter l Skills Prctice

23 Nme Dte Construct ech isosceles tringle described using the given segment. 1. Construct right isosceles tringle ABC with segment BC s the hypotenuse by constructing 5 ngles t B nd C. B C A B C. Construct right isosceles tringle WXY with segment WX s the hypotenuse by constructing 5 ngles t W nd X. W X W X X 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 53

24 3. Construct right isosceles tringle PQR with RQ s leg nd R s the right ngle. R Q P R Q. Construct right isosceles tringle DEF with DF s leg nd D s the right ngle. D F E 010 Crnegie Lerning, Inc. D F 5 Chpter l Skills Prctice

25 Skills Prctice Skills Prctice for Lesson. Nme Dte Other Specil Right Tringles Properties of Tringle Vocbulry Write the term tht best completes ech sttement. 1. A(n) 30º 60º 90º tringle is formed by dividing n equilterl tringle in hlf by its ltitude.. The 30º 60º 90º Tringle Theorem sttes tht the length of the hypotenuse in tringle is two times the length of the shorter leg, nd the length of the longer leg is 3 times the length of the shorter leg. Problem Set Determine the mesure of the indicted interior ngle. 1. A. D 30 B C E G F m ABC 60º m DFE 60º 3. H 30. R 60 J A K S A T m HAK 90º m TRA 30º 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 55

26 Given the length of the short leg of tringle, determine the lengths of the long leg nd hypotenuse. Write your nswers s rdicls in simplest form ft 60 c in. c b 30 b 30 3 ft 5 in. b 3 3 ft b 5 3 in. c (3) 6 ft c (5) 10 in mm b c cm 6 mm 15 cm b mm b cm c 6 mm c 15 cm Given the length of the hypotenuse of tringle, determine the lengths of the two legs. Write your nswers s rdicls in simplest form. 60 c b m km b 30 b Crnegie Lerning, Inc. 11. c 0 m c 16 km 0 10 m 16 8 km b 10 3 m b 8 3 km yd b ft c 6 3 yd c ft yd ft b ( 3 3 ) yd b ( ) 3 6 ft b Chpter l Skills Prctice

27 Nme Dte Given the length of the long side of tringle, determine the lengths of the short leg nd hypotenuse. Write your nswers s rdicls in simplest form c 8 3 in c 11 3 m b 8 3 in. b 11 3 m in m 3 c (8) 16 in. c (11) m c c 30 1 mi ft b 1 mi b 18 ft mi c ( 3 ) 8 3 mi b ( 6 3 ) 1 3 ft 6 3 ft Determine the re of ech tringle. Round your nswer to the nerest tenth, if necessry cm b cm b 3 3 cm A A cm The re of the tringle is pproximtely 7.8 squre centimeters. 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 57

28 km b km b 6 3 km A A 36 3 A km The re of the tringle is pproximtely 31. squre kilometers. 19. Universl Sporting Goods sells pennnts in the shpe of 30º 60º 90º tringles. The length of the longest side of ech pennnt is 16 inches. c 16 in in. b 8 3 in. A A 6 3 A in. The re of the pennnt is pproximtely 55. squre inches. 010 Crnegie Lerning, Inc. 0. A fctory produces solid drfting tringles in the shpe of 30º 60º 90º tringles. The length of the side opposite the right ngle is 15 centimeters. c 15 cm 15 cm b 15 ( 3 ) 15 3 cm A A cm 8 The re of the drfting tringle is pproximtely 8.7 squre centimeters. 58 Chpter l Skills Prctice

29 Nme Dte Construct ech tringle described using the given segment. 1. Construct tringle by first constructing n equilterl tringle with MN s side, nd then bisecting one of the sides. M N M N. Construct tringle RST by first constructing n equilterl tringle with RS s side, nd then bisecting the ngle t R. R S R T S 010 Crnegie Lerning, Inc. Chpter Skills Prctice 59

30 3. Construct tringle EFG with EF s the side opposite the 30 ngle by first constructing n equilterl tringle. E F F G E. Construct tringle ABC by first copying ngle A, nd then drwing AB s the hypotenuse. A B A 30 C 010 Crnegie Lerning, Inc. A B 60 Chpter l Skills Prctice

31 Skills Prctice Skills Prctice for Lesson.5 Nme Dte Pst Anyone? Tringle Inequlity Theorem Vocbulry Identify n exmple of ech term in the digrm of tringle ABC. 1. Tringle Inequlity Theorem B AB BC AC. uxiliry line Line BD is n uxiliry line. A D C Problem Set Without mesuring the ngles, list the ngles of ech tringle in order from lest to gretest mesure in. F G 11 in. 9 in. H..7 cm Y 3.6 cm X.1 cm W The smllest ngle of tringle is opposite the shortest side. So, the ngles from lest to gretest re H, F, G. The smllest ngle of tringle is opposite the shortest side. So, the ngles from lest to gretest re Y, X, W. 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 61

32 3. Q. T 1 in S 8 ft ft 9 in 15 in P 6.3 ft R U The smllest ngle of tringle is opposite the shortest side. So, the ngles from lest to gretest re P, Q, R. 5. F The smllest ngle of tringle is opposite the shortest side. So, the ngles from lest to gretest re S, U, T. 6. K.6 yd 9. yd 5.8 m. m E 6 yd G M 5. m L The smllest ngle of tringle is opposite the shortest side. So, the ngles from lest to gretest re G, F, E. The smllest ngle of tringle is opposite the shortest side. So, the ngles from lest to gretest re M, K, L. Determine whether it is possible to form tringle using ech set of segments with the given mesurements. Explin your resoning in.,.9 in., 5 in ft, 9 ft, 11 ft Yes. Becuse , nd Yes. Becuse , nd 5.9 is greter thn is greter thn Crnegie Lerning, Inc. 9. m, 5.1 m, 1.5 m cm, 8.1 cm, 9.8 cm No. Becuse , nd Yes. Becuse , nd 9.1 is not greter thn is greter thn yd, 5 yd, 1 yd km, 6.3 km, 7.5 km No. Becuse , nd No. Becuse , nd 15 is not greter thn is not greter thn mm, 300 mm, 190 mm in., 11 in., 8. in. Yes. Becuse , nd No. Becuse , nd 30 is greter thn is not greter thn Chpter l Skills Prctice

33 Nme Dte cm, 1 cm, 17 cm ft, 8 ft, 8 ft No. Becuse , nd Yes. Becuse , nd 9 is not greter thn is greter thn 8. Write n inequlity tht expresses the possible lengths of the unknown side of ech tringle. 17. Wht could be the length of AB? 18. Wht could be the length of DE? A D 10 m 6 cm B 8 m C AB AC BC AB 10 m 8 m AB 18 m 19. Wht could be the length of HI? 0 in. I F 9 cm E DE DF EF DE 6 cm 9 cm DE 15 cm 0. Wht could be the length of J L? 1 ft J H 1 in. G HI GH GI HI 1 in. 0 in. HI 3 in. K 7 ft L JL JK KL JL 1 ft 7 ft JL 19 ft 010 Crnegie Lerning, Inc. Chpter l Skills Prctice 63

34 1. Wht could be the length of MN? M. Wht could be the length of QR? P N 11 cm O 3 cm 9 mm 13 mm R Q MN NO MO MN 11 cm 3 cm MN 1 cm QR PR PQ QR 9 mm 13 mm QR mm 010 Crnegie Lerning, Inc. 6 Chpter l Skills Prctice

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