Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior ngles. interior exterior If we extend one of the sides, it forms n ngle outside of the tringle, lled n exterior ngle. n exterior ngle is supplementry to its djent interior ngle. Eh interior ngle is lso referred to s the inluded ngle etween the two sides tht rete it. For exmple, is the inluded ngle etween sides nd. Eh point,, nd, is lled vertex, nd together they re lled verties (the plurl of vertex). Wheres the leling of the verties is usully ritrry (, nd n go in ny position round the tringle), leling the sides s, nd is not ritrry. For exmple, side must go opposite, nd so on. property of the sides of tringle is tht the sum of the lengths of ny two sides is greter thn the length of the third side. This is espeilly true for the longest side: + > nd < + + > nd < + + > nd < + the length of the shortest side + the length of the middle side > the length of the longest side Exmple 1: n these e the lengths of the sides of tringle? ssume ll mesures re in inhes. ) 5, 8, 10 ) 3, 4, 8 ) 5, 7, 12 nswer: ) Yes euse the sum of the lengths of the shorter legs, 5 + 8 = 13, is greter thn the longest side, 10. ) No, euse 3 + 4 is not greter thn 8. ) No, euse 5 + 7 is not greter thn 12. Note: visul nswer is on the next pge. Se 1.3 Tringles 21 Roert H. Prior, 2016 opying is prohiited.
22 Se 1.3 Tringles Here is visul nswer for the three sets of side mesures in Exmple 1: ) ) ) 5 5 10 Yes, these mesures form tringle. 8 8 4 4 No, these mesures do not form tringle. 8 3 3 5 5 12 No, these mesures do not form tringle. 7 7 Strting with the longest side, drw it horizontlly. Then, think of the other two side mesures strting t either end of the longest side onneted y swinging hinge. If the two smller sides n swing wy from the longest side nd meet somewhere, tringle n e formed, s in ). Otherwise, no tringle n e formed, s in ) nd ). LSSIFITION OF TRINGLES Y SIDE MESURES Isoseles Tringle: i) Two sides re ongruent to eh other; Equilterl Tringle: i) ll three sides re ongruent to eh other; ii) the two ngles, nd, opposite the ongruent sides re lled se ngles nd re ongruent to eh other. ii) ll three ngles re ongruent to eh other. Is n equilterl tringle lso isoseles? Isoseles Tringle: n isoseles tringle n e oriented differently thn the one presented ove. In this se, the ongruent ngles re still lled se ngles. Slene Tringle: i) No two sides re ongruent to eh other; ii) likewise, no two ngles re ongruent to eh other. Se 1.3 Tringles 22 Roert H. Prior, 2016 opying is prohiited.
Se 1.3 Tringles 23 LSSIFITION OF TRINGLES Y NGLE MESURES Right Tringle: tringle with one right ngle. (It is not possile for tringle to hve two right ngles.) In right tringle, the side opposite the right ngle is lled the hypotenuse. The other two sides re lled legs of the right tringle. Only right tringles hve hypotenuses, nd the hypotenuse is lwys the longest side in right tringle. leg leg hypotenuse Otuse Tringle: tringle in whih one ngle is greter thn 90. ute Tringle: tringle in whih ll three ngles re less thn 90. Olique Tringle: tringle in whih no ngle is right ngle. Otuse tringles nd ute tringles re oth exmples of olique tringles. THE SUM OF THE NGLES IN TRINGLE The sum of the interior ngles in ny tringle is 180. This mens tht m 1 + m 2 + m 3 = 180. 2 nd tht m + m + m = 180. 1 3 Se 1.3 Tringles 23 Roert H. Prior, 2016 opying is prohiited.
24 Se 1.3 Tringles Proof: To show tht the sum of the ngles in tringle is 180 we n use two prllel lines nd two trnsversls: line 1 line 1 4 2 5 line 2 line 2 1 3 trnsversl 1 trnsversl 2 trnsversl 1 trnsversl 2 t point, three djent ngles form line 1, so their sum must e tht of stright ngle, 180 : m 4 + m 2 + m 5 = 180 lso, 1 nd 4 re lternte interior ngles formed y the prllel lines nd trnsversl 1, so Likewise, 3 nd 5 re lternte interior ngles formed y the prllel lines nd trnsversl 2, so 1 4 whih mens m 1 = m 4. 3 5 whih mens m 3 = m 5. Repling 4 with 1 nd 5 with 3 we get m 1 + m 2 + m 3 = 180 Hene, the sum of the (interior) ngles in the tringle is 180. This is true for every tringle. THE UTE NGLES IN RIGHT TRINGLE In every tringle, t lest two of the ngles re ute. This is true for right tringles, otuse tringles, nd ute tringles. In right tringle, the two ute ngles re omplementry. This is esy to demonstrte: m + m + m = 180 90 + m + m = 180 m + m = 90 So, nd re omplementry.! nd! re omplementry. Se 1.3 Tringles 24 Roert H. Prior, 2016 opying is prohiited.
Se 1.3 Tringles 25 Exmple 2: onsider. Given the mesures of two of the ngles, find the mesure of the third ngle. ) m = 25 nd m = 71. Find m. ) m = 103 15 42 nd m = 22 48 35. Find m. nswer: dd the two known ngles nd sutrt their sum from 180. ) 25 + 71 = 96 ; m = 180 96 = 84 ) 103 15 42 + 22 48 35 djust the seonds: djust the minutes: 125 63 77 125 64 17 126 04 17 Mke 1 into 60 Mke 1 into 60 Now sutrt 180 00 00 179 60 00 179 59 60 126 04 17 126 04 17 126 04 17 m = 53 55 43 SIDES OPPOSITE NGLES In ny tringle, the shortest side is lwys opposite the smllest ngle. Likewise, the longest side is lwys opposite the lrgest ngle. This is why the ngles in n equilterl tringle re ll the sme mesure: there is no one side longer thn nother, so there is no ngle lrger thn nother. Likewise, this is why the se ngles of n isoseles tringle re ongruent: the sides opposite the se ngles re the sme length, neither is longer nor shorter thn the other, so the se ngles must lso e the sme mesure. Exmple 3: List the sides in order from shortest to longest. Proedure: First, m = 180 (75 + 40 ) = 65. nswer: Next, the shortest side is opposite the smllest ngle, nd so on. The lengths of the sides from shortest to longest 75 40 re,,. Se 1.3 Tringles 25 Roert H. Prior, 2016 opying is prohiited.
26 Se 1.3 Tringles Exmple 4: Proedure: nswer: List the ngles in order from smllest to lrgest. The smllest ngle is opposite the shortest side, nd so on. The size of the ngles from smllest to lrgest Y 15 18 11 Z re Y, Z, X. X THE PYTHGOREN THEOREM In ny right tringle, with m = 90, the lengths of the sides hve the fmilir Pythgoren reltionship: leg 2 + leg 2 = hypotenuse 2 2 + 2 = 2 Exmple 5: In right tringle, given the mesures of two sides, find the length of the third side. Here, is the length of the hypotenuse nd nd re lengths of legs. Simplify. ) = 13 in. nd = 12 in. Find. ) = 7 ft nd = 2 ft. Find. ) = 5 km nd = 3 km. Find. d) = 3 2 m nd = 4 3 m. Find. nswer: Use the Pythgoren Theorem to find the missing side vlue. Even though the solving of eh eqution leds to two solutions, one positive nd one negtive, we use only positive solutions for lengths nd distnes. ) 12 2 + 2 = 13 2 ) 2 + 2 2 = ( 7 ) 2 144 + 2 = 169 2 + 4 = 7 2 = 25 2 = 3 = ± 25 = ± 3 = 5 in. = 3 ft ) ( 5 ) 2 + ( 3 ) 2 = 2 d) ( 3 2 ) 2 + ( 4 3 ) 2 = 2 5 + 3 = 2 9 2 + 16 3 = 2 8 = 2 18 + 48 = 2 ± 8 = 66 = 2 = 2 2 km ± 66 = = 66 m Se 1.3 Tringles 26 Roert H. Prior, 2016 opying is prohiited.
Se 1.3 Tringles 27 PYTHGOREN TRIPLES ny set of three positive integers,, nd tht hve suh Pythgoren reltionship is lled Pythgoren triple nd n e written (,, ). The set of numers (3, 4, 5) is one suh Pythgoren triple. s suh, tringle tht hs side mesures (in inhes or feet or meters) of 3, 4 nd 5 is right tringle. The longest mesure is the length of the hypotenuse: 3 2 + 4 2 = 5 2 9 + 16 = 25 whih is true! Note: It is true tht 3, 7 nd 4 n e the sides of right tringle, euse 3 2 + ( 7 ) 2 = 4 2. However, these numers do not form Pythgoren triple euse not ll of them re integers. Generting Pythgoren Triples For ny two positive integers, m nd n where m > n, we n generte Pythgoren triple y pplying the following formuls: = m 2 n 2 = 2mn = m 2 + n 2 For exmple, if m = 4 nd n = 3, then = 4 2 3 2 = 2 4 3 = 4 2 + 3 2 = 16 9 = 24 = 16 + 9 = 7 = 24 = 25 So, (7, 24, 25) is Pythgoren triple: 7 2 + 24 2 = 25 2 49 + 576 = 625 625 = 625 True! Here is why these reltionships generte Pythgoren triples: 2 + 2 = 2 (m 2 n 2 ) 2 + (2mn) 2? = (m 2 + n 2 ) 2 m 4 2m 2 n 2 + n 4 + 4m 2 n 2 m 4 2m 2 n 2 + 4m 2 n 2 + n 4? = m 4 + 2m 2 n 2 + n 4? = m 4 + 2m 2 n 2 + n 4 m 4 + 2m 2 n 2 + n 4 = m 4 + 2m 2 n 2 + n 4 Se 1.3 Tringles 27 Roert H. Prior, 2016 opying is prohiited.
28 Se 1.3 Tringles Setion 1.3 Fous Exerises n these e the lengths of the sides of tringle? ssume ll mesures re in inhes. 1. 16, 12, 8 2. 18, 32, 14 3. 61, 28, 32 4. 47.8, 16.8, 32.7 5. 4 3, 5 3, 2 3 6. 18, 8, 32 In, given the mesures of nd, find the mesure of. lso, identify the type of tringle it is sed on its ngle mesures, either ute, right, or otuse. 7. m = 63 nd m = 48 8. m = 13.75 nd m = 76.25 9. m = 36 15 nd m = 94 53 10. m = 22 49 35 nd m = 41 18 52 11. Is it possile for right tringle to hve the following desription? If not, why not? ) Equilterl ) Isoseles ) Slene d) Olique 12. Drw n isoseles right tringle nd lel the ngle mesures. Se 1.3 Tringles 28 Roert H. Prior, 2016 opying is prohiited.
Se 1.3 Tringles 29 XYZ is n isoseles tringle. X nd Y re the ongruent se ngles. Given one of the ngle mesures, find the mesures of the other two. 13. m X = 48 14. m Y = 76 15. m Z = 34 16. m Z = 105 17. m X = 53 37 49 18. m Z = 102 24 06 sed on the given digrm, list the sides in order from shortest to longest. 19. 55 20. 73 E f 41 d 68 F e D sed on the given digrm, list the ngles in order from smllest to lrgest. 21. H 23 G 29 27 F 22. 7.8 P Q 5.3 9.2 R Se 1.3 Tringles 29 Roert H. Prior, 2016 opying is prohiited.
30 Se 1.3 Tringles In right, with m = 90, given the lengths of two of the sides, find the length of the third side. 23. = 6, = 4 24. = 2, = 3 25. = 3, = 3 26. = 8, = 4 27. = 5 2, = 5 28. = 4 3, = 6 29. = 2 6, = 3 30. = 3 2, = 1 2 Generte Pythgoren triples using the following vlues of m nd n; Use = m 2 n 2, = 2mn, nd = m 2 + n 2 31. m = 2 nd n = 1 32. m = 3 nd n = 1 33. m = 5 nd n = 2 Se 1.3 Tringles 30 Roert H. Prior, 2016 opying is prohiited.