Tag: ângulos

0

[PQRS] é um paralelogramo

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 111 Ex. 6

Enunciado

[PQRS] é um paralelogramo.

  1. Quantos triângulos estão representados na figura?
     
  2. Calcula:   
  • $P\hat{Q}R$
     
  • $S\hat{T}R$
     
  • $P\hat{S}R$
     
  • $Q\hat{T}R$

Resolução >> Resolução

  1. Na figura estão representados 8 triângulos: [PQT], [QTR], [RTS], [STP], [PQR], [RPS], [RSQ] e [PQS].
     
  2. (Vai anotando na figura as amplitudes calculadas)
  • $P\hat{Q}R=180{}^\text{o}-(Q\hat{P}R+Q\hat{R}P)=180{}^\text{o}-(18{}^\text{o}+30{}^\text{o})=132{}^\text{o}$, pois a soma das amplitudes dos três ângulos internos de um triângulo é 180º.
Constrói um paralelogramo [MNPQ] 0

Constrói um paralelogramo [MNPQ]

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 111 Ex. 2

Enunciado

Constrói um paralelogramo [MNPQ], sabendo que $\overline{MN}=10\,cm$, $\overline{MQ}=5,4\,cm$ e $\hat{M}=60{}^\text{o}$.

A seguir, traça as suas diagonais e designa por O o seu ponto de intersecção.

Determina:

  1. a amplitude do ângulo interno P;
  2. a amplitude do ângulo interno Q;
  3. o perímetro do paralelogramo.

Resolução >> Resolução

Reproduz a construção:

var parameters = { "id": "ggbApplet", "width":895, "height":475, "showMenuBar":false, "showAlgebraInput":false, "showToolBar":false, "customToolBar":"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6", "showToolBarHelp":false, "showResetIcon":true, "enableLabelDrags":false, "enableShiftDragZoom":false, "enableRightClick":false, "errorDialogsActive":false, "useBrowserForJS":false, "preventFocus":false, "language":"pt", // use this instead of ggbBase64 to load a material from GeoGebraTube // "material_id":12345, "ggbBase64":"UEsDBBQACAgIAN22HkcAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiuBQBQSwcI1je9uRkAAAAXAAAAUEsDBBQACAgIAN22HkcAAAAAAAAAAAAAAAAXAAAAZ2VvZ2VicmFfZGVmYXVsdHMyZC54bWztml9T4zYQwJ/vPoXGT+0DieXESWAIN9zNdMoMx3UKc9NXxd44KrLkWjJx8ulPlvwvkNBgODLQvmCtIsmr3+5KK5nTT3nM0B2kkgo+dXDPdRDwQISUR1MnU/OjifPp7ONpBCKCWUrQXKQxUVPHL1rW/bTUw8NBUYdySU+4uCIxyIQEcB0sICaXIiDKNF0olZz0+8vlslcN2hNp1I8i1ctl6CCtEJdTpyyc6OE2Oi0Hprnnurj/19dLO/wR5VIRHoCDtLIhzEnGlNRFYBADV0itEpg6iWCrSHAHMTIDNnX+qOSyx9QZu87Zxw+njHK4VisGSC1ocMtBao08pxzGtYXfaRhCAc3pF33kQiyRmP0NgR5HpRnUrzGCaaN//iKYSFGqu/kDB2nIPnbQzAxKWLIgutQrR2RkBSm6I6z4tazRA34VIdjaoa0lnMaGLpIKkkIhJBOA0JRqlRM9nLHqnDBp9Dntl3i2gioYbJCyFQ0q/GqoXAPKfcDJPTSnecaDYsCr7ySt58AzxlqcRr7TZc6e7++Y9dg/9LQTQblq+YaW0C/zFODX1ryx22nebVsbBj/R2njbtD+cBkKkoUT51LkiVw5alc+1fZomhsA1XZevHLRrTTA0+j0RYwgJcB0saoMl7sRyNDEwi8fMPt4vTEZlw/LSCA2+wRZftDru44zYvR+ER/i11p5uC+x+RI/wk/3zW3uzxF4nr8SeXdnM8z8Z5Rf8T4joRuKBB/+z7MRy0yOH73jPMU0sK1n8nTqBiBMG+QsClhAVUs3rupJrxF63rejAKdxegLustCJTrHjXBVf6MAQmG5RW5dbLbwGSG935G79JCZfFIcq2qWA9tq+10vDLzRTce36K9Z5sAf/wjfCgOjpoQNW/ABZBJhvCVqoRT94oYpLllFGSrh744tPJPu/843Xb2Xavyd7Bzz8pWT22QnY78B3cZd7qClk54U4HfH5ScBB7vGSg3ulZiyZEv5dizWjbAektMPpJPrsl1SKpAkkJf5yzgrxJnm6M0LoQOSzkHTvC7sloo0SNchdWat1J2OnMqabESaw72BdR/pkEt1EqMh4+iPOXmfyrHb93wwkEp0Gt/Bcr1XCGbzSeOqVdNAJuFxiJUO6WnxFWrtUcrauaHJc1K1zWrHHLllrllObovOp3XjU/96rCoCoMq4LfwtMt/zOGTHR4t7b0e6vjsNuZ5/A3/O/YoK+QWPAshrQV5FeVXDuGb8Ncj5dV5+tK933Cuvocwmio3SCm2gRHOtONid7Piox3JgXLFFwHKQBvPqFZ11vSUC2KM6DhlleWKJ9zmhfuYZsuRErXgiuy4apdXOO+IxZzeO5KSnjEmlA6t1KD2F4ymkb37zG2k2/jdEuao543GeCJP3DHeHzsT0Z70sWTrnRf7K75yYvFk+zqlXZNg9bVkbvL2O5k7I1Gw5HnHx+P8Wg4frEvaDWc3+qK5gvae9pMB90S+JkQDEiD6XMlt27jHyxGu/Ku/d3x2fSCBQS3M5FvhMy9mfZbH+z71T8FnP0AUEsHCD5gRIp7BAAAmyAAAFBLAwQUAAgICADdth5HAAAAAAAAAAAAAAAAFwAAAGdlb2dlYnJhX2RlZmF1bHRzM2QueG1s7VbRbtsgFH1evwLx3tiO47ap4lZR97BJbbWpL3sl+MZhw+ACSZz+2v5h3zTAJnWatdJSqdq0vdiHy73XcM7lmsllU3G0AqWZFDlOBjFGIKgsmChzvDTz4zN8eXE0KUGWMFMEzaWqiMlx5jy3cXY0SEaps6FGs3Mhb0kFuiYU7ugCKnItKTHedWFMfR5F6/V6EJIOpCqjsjSDRhcY2QUJneMOnNt0O0Hr1LsP4ziJvtxct+mPmdCGCAoY2cUWMCdLbrSFwKECYZDZ1JBj0jCd2k9wMgOe46kbvseo889xmsQpvjh6N9ELuUZy9hWotRq1hG2MH0TOx05fSS4VUjm2+y79c+afhNcLYpHlw7tysgGFVoS72c5is93IAlrrqLUSwSpPE9IGaisHRroGKDxqt2Cz1zadl2dOuO4Ww5mAO7PhgMyC0W8CtKVw2Aty4AMrCnAqtzFwL9oQ7Z45romyohnFqP1Gi8Hu7cd35z6JOir3SLXLEdBj9ZMf79BqxTqI1vHY8zpMxp5Z/95ym70Vt1RKVWjUtIKiTfd+6N7rntBz4g5Ot5pB8jJxVApGe8R9FJZvbblxi6RLtYKd0swO43CYZZ7EZHi6V57JH12erASxstuUStuuEnfdaRMH/oOlSYIySWd56IDPY5esWIOmIW4a3KfDANIARgFkPVGfnhNW1ZxRZg7d2vMVcb8khT9+naKfw/ixDNI4eVUZ7Peo0zc7SK9RAk1PAjgN4CyA8VatF9qU5JsFFEqKx07VM/UZbg/aITX7u6okWepVyZI9WUZvo8oL7cl1IEqUAc2I6PWpKzfx9L958q/8N58nTIDZbvfW4X5NZf9ryrrrpZrbO+Gvqqqb2mVt9Jf2uj4DUe86GoUr78VPUEsHCBS5/A+XAgAAeQsAAFBLAwQUAAgICADdth5HAAAAAAAAAAAAAAAADAAAAGdlb2dlYnJhLnhtbO1b3W7byBW+3n2KgS56UVj0/JNM7V04u1g0QOx412lRtCgKihrJXFOklqRsOdhXKtC+QnvfZ+o5MyRF/UaSnfUGrRNlOMMzZ+ac7/zNyDn7ej5Jyb0pyiTPznvMoz1isjgfJtn4vDerRv2g9/VXX56NTT42gyIio7yYRNV5TyFlOw96HpMCx5IhsImpoHww6tM40H0pZdgfBCPdj3wdqWikWCgGPULmZfIqy6+iiSmnUWxu4lszid7mcVRZprdVNX11evrw8OA1y3t5MT4djwfevBz2CGw9K8979cMrYLc06UFYck4pO/3T5VvHvp9kZRVlsekRFGuWfPXlF2cPSTbMH8hDMqxuz3shEz1ya5LxLcipedgjp0g0BWGnJq6Se1PC1E7XylxNpj1LFmX4/gv3RNJWnB4ZJvfJ0BTnPerxgIpQCgqqoRpWyIvEZFVNyeoVTxteZ/eJeXBM8cmuJ2noAwJJmQxSc94bRWkJMiXZqAB9wnaKGXTL6jE1g6ho+ovdsBP7B0iSDwa5AXRODdDh9ERwceJTeqIUdbvpLK0Y75Eqz1PLmZKfCSOKwoewkJwQ7cMIJ0wRCSMBjPhE4JhikgiCJEwQKaGVOMw0vlMwX1HCGAwTTgnnhDPCBXSVIkoT5eNEDrQ6tMwofJAatgMfgWNCwMeOCQkfjk/ASDk2sAkltH1SSA38Fcft20EREBnCQjigfEYE7AH6PiXAUSB7ZoWQlOBfRiSy5z7hAQF+IDdypnwHKHV/gUo9sAJLA4rqgsIADPxo+Fi0VkCRy5AAAhRkO8GGuQa3q7V7Rd0YFa7hrpGuUY5GuunSkTppqXQ0UjxVzEZI3hWSnljhNgoYdARkKAAAgju3jSC4Z2b3jo2su9p1rZlRRuvRAP8JsQP60IF9eKI8opFHHAIa66zqPHT7omse3KwYhGo/DT7NNMVWxPg26XYpdTVAreu0WY+pznoKQhL+tZ+1FcUuET8aEo9YUC+53S8trn/IikeLe3bapJ+zWlRS3iJtbbGVmZQYc0TYZgKNsbpOBz7vpIMTTAhaLXICZoRgKSeooJMYICtoHPRtloE1MKy7JMFlkydO6kzx81qmgMAuF7EdtoasMHLUwR1W593wziEccOJjVIRchZGBcGDJCWQFjfO2RP4emeZl0ur11qTTFhCrwiSbzqoltcWTYfNY5UAdpbbCqemHeXz3ulV0zclEZdVlCwXCoghxBcNSjfLFWRoNTAql3A1aASH3UYp+bFcY5VlFGguQbmxcRNPbJC5vTFXBrJL8GN1Hb6PKzL8D6rJZ29LGeVZeF3n1TZ7OJllJSJyntBUuT1nnmXeeRSsBdGTnheq+0J0X/sZ1c3hDZqWB9fOibMij4fANUiwCGijwXZY+vi5MdDfNk21iXEX3ydh6Bqq9js3TNHp8PasqhNXxx5FvDfwDYRCr3SlMXaZAhjeVAUT7rGdBskXmmZnFaTJMouyP4ExNTXc1mwxMQexjjkvaPSEopKlGbXRvqlHJWLPtvBjePJbge2T+Z1PAZC6lR5mkMtA0FL70Yd5j/YqFnoAfHmifBb5C148jDBoi9Pww9Dk4u/3BOZtfMYjmdmlz35pHNDcLTY8L1Fqn86Z8naeLIav8b6JpNSvs4QLySYFSXWTj1FgDtTqHKj2+G+TzG2eZwvF6/zg1GGDtDgZjCzqBoMYV7HhctwPXWhrcWktFLQ21FLQx9WTYvmchtxS2HbjWUoHvuK3VorJGTEabZZLSWgvtLfuq9Tws+mdZUr1tOlUS3y1ExQnOAhr7XebJnovn2emK9Z3dmSIzae0AAOYsn5UuPHR8A7zrOqpuL7LhD2YMnnEdYWapgLUjXWx5aOJkAhPdeK28CIH9A2zVjQ7NuDCNiC7YOdXat7Rr12vDltV3RT55k92/B6tZ2erZaSPPWRkXyRStkwwg1d2Zhf0NkzKCRDnszltSi/h2i2dR9PTHzvMH99xnnmo9Sdk3c2vNWCNZurrX19jd7D0uajyT86y5yrp91mXAc5rn87Hkz8YSAnVlusz2jhxgEdMpGhCYf1tFdTZVp8x6mSL/EfNtnpFqofcVf0PDsokGGNS0SYXbh3w1q27zwl4OwH6hRaNMzcRAcnYMLfKtKi7tHQNuh+QDXHlFVa5j7vE8aDcIVBvDoBU7Sqe3kbNpF+6iR8xGHcezbN+NRqWpyBwMmQNCjxggO68v8+G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// is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View var views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0}; var applet = new GGBApplet(parameters, '5.0', views); window.onload = function() {applet.inject('ggbApplet')};

Explica a construção.…

Constrói um paralelogramo [MATE] 0

Constrói um paralelogramo [MATE]

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 110 Ex. 8

Enunciado

Constrói um paralelogramo [MATE], tal que $\overline{MA}=5\,cm$, $\overline{AT}=2,5\,cm$ e $\hat{A}=55{}^\text{o}$.

Resolução >> Resolução

Reproduz a construção:

var parameters = { "id": "ggbApplet", "width":834, "height":362, "showMenuBar":false, "showAlgebraInput":false, "showToolBar":false, "customToolBar":"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6", "showToolBarHelp":false, "showResetIcon":true, "enableLabelDrags":false, "enableShiftDragZoom":false, "enableRightClick":false, "errorDialogsActive":false, "useBrowserForJS":false, "preventFocus":false, "language":"pt", // use this instead of ggbBase64 to load a material from GeoGebraTube // "material_id":12345, "ggbBase64":"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"}; // is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View var views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0}; var applet = new GGBApplet(parameters, '5.0', views); window.onload = function() {applet.inject('ggbApplet')};

Explica a construção.…

Constrói um paralelogramo [DOCE] 2

Constrói um paralelogramo [DOCE]

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 110 Ex. 7

Enunciado

Constrói um paralelogramo [DOCE], tal que $\overline{DO}=4\,cm$, $\overline{CO}=3\,cm$ e $\overline{DC}=6\,cm$.

Resolução >> Resolução

Reproduz a construção:

var parameters = { "id": "ggbApplet", "width":905, "height":449, "showMenuBar":false, "showAlgebraInput":false, "showToolBar":false, "customToolBar":"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6", "showToolBarHelp":false, "showResetIcon":true, "enableLabelDrags":false, "enableShiftDragZoom":false, "enableRightClick":false, "errorDialogsActive":false, "useBrowserForJS":false, "preventFocus":false, "language":"pt", // use this instead of ggbBase64 to load a material from GeoGebraTube // "material_id":12345, "ggbBase64":"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"}; // is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View var views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0}; var applet = new GGBApplet(parameters, '5.0', views); window.onload = function() {applet.inject('ggbApplet')};

Explica a construção.…

Soma dos ângulos de um quadrilátero 0

Soma dos ângulos de um quadrilátero

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 110 Ex. 3

Enunciado

Prova que a soma das amplitudes dos ângulos internos de um quadrilátero é 360º.

Resolução >> Resolução

Consideremos o quadrilátero [ABCD]:

var parameters = { "id": "ggbApplet", "width":740, "height":398, "showMenuBar":false, "showAlgebraInput":false, "showToolBar":false, "customToolBar":"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6", "showToolBarHelp":false, "showResetIcon":true, "enableLabelDrags":false, "enableShiftDragZoom":false, "enableRightClick":false, "errorDialogsActive":false, "useBrowserForJS":false, "preventFocus":false, "language":"pt", // use this instead of ggbBase64 to load a material from GeoGebraTube // "material_id":12345, "ggbBase64":"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// is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View var views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0}; var applet = new GGBApplet(parameters, '5.0', views); window.onload = function() {applet.inject('ggbApplet')};

Tracemos a diagonal [AC].…

A medida da amplitude do ângulo externo 0

A medida da amplitude do ângulo externo

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 103 Ex. 6

Enunciado

A medida da amplitude do ângulo externo em B, no triângulo [ABC], é 100º.

Sabendo que $\hat{B}=\hat{C}$:

  1. determina a medida da amplitude de cada um dos ângulos internos do triângulo;
  2. indica qual o lado de maior comprimento do triângulo e o de menor comprimento. Justifica.

Resolução >> Resolução

  1. var parameters = { "id": "ggbApplet", "width":278, "height":313, "showMenuBar":false, "showAlgebraInput":false, "showToolBar":false, "customToolBar":"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6", "showToolBarHelp":false, "showResetIcon":true, "enableLabelDrags":false, "enableShiftDragZoom":false, "enableRightClick":false, "errorDialogsActive":false, "useBrowserForJS":false, "preventFocus":false, "language":"pt", // use this instead of ggbBase64 to load a material from GeoGebraTube // "material_id":12345, 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// is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View var views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0}; var applet = new GGBApplet(parameters, '5.0', views); window.onload = function() {applet.inject('ggbApplet')}; Como os ângulos ABC e CBD são suplementares, então $A\hat{B}C=180{}^\text{o}-C\hat{B}D=180{}^\text{o}-100{}^\text{o}=80{}^\text{o}$.
0

Sabendo que…

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 103 Ex. 5

Enunciado

Sabendo que $\hat{B}=62{}^\text{o}$, $\overline{AB}=\overline{BC}$ e $\overline{CD}=\overline{CE}$ , calcula $C\hat{D}E$.

 

Resolução >> Resolução

Como sabemos, num triângulo, a lados geometricamente iguais, opõem-se ângulos geometricamente iguais.

Logo, no triângulo [ABC], são geometricamente iguais os ângulos BAC e BCA, pois opõem-se a lados geometricamente iguais.

Por outro lado, sabemos que a soma das amplitudes dos ângulos internos de um triângulo é 180º.…

0

Observa a figura

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 103 Ex. 4

Enunciado

Observa a figura onde [MN] é paralelo a [BC].

Calcula:

  1. $M\hat{A}N$
  2. $A\hat{B}D$

Resolução >> Resolução

  1. O ângulo MNA é suplementar do ângulo assinalado com 142º de amplitude, pois são ângulos de lados paralelos, sendo um agudo e outro obtuso. Assim, $M\hat{N}A=180{}^\text{o}-142{}^\text{o}=38{}^\text{o}$.
     
    Como a soma das amplitudes dos ângulos internos dum triângulo é 180º, temos: $M\hat{A}N=180{}^\text{o}-(A\hat{M}N+M\hat{N}A)=180{}^\text{o}-(70{}^\text{o}+38{}^\text{o})=72{}^\text{o}$.
0

Calcula o valor de x em cada figura

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 103 Ex. 2

Enunciado

Calcula o valor de x em cada figura, considerando r//s.

Resolução >> Resolução

FIGURA a)

Os ângulos considerados são geometricamente iguais, pois são ambos agudos e de lados paralelos.
Logo, temos:

$\begin{array}{*{35}{l}}
   2x-95=25 & \Leftrightarrow  & 2x=120  \\
   {} & \Leftrightarrow  & x=60  \\
\end{array}$

Portanto, $x=60{}^\text{o}$.

FIGURA b)

Os ângulos considerados são suplementares pois, sendo um agudo e outro obtuso, possuem lados paralelos:
Logo, temos:

$\begin{array}{*{35}{l}}
   (5x+30)+(x+6)=180 & \Leftrightarrow  & 5x+x=180-30-6  \\
   {} & \Leftrightarrow  & 6x=144  \\
   {} & \Leftrightarrow  & x=24  \\
\end{array}$

Portanto, $x=24{}^\text{o}$.…

0

Um triângulo isósceles

Do espaço ao plano: Matematicamente Falando 7 - Parte 2 Pág. 102 Ex. 3

Enunciado

No triângulo isósceles [MAR], $\overline{RA}=\overline{MA}$ e $\hat{A}=50{}^\text{o}$.

Determina ${\hat{R}}$ e ${\hat{M}}$.

Resolução >> Resolução

Como a soma dos ângulos internos de um triângulo é um ângulo raso, vem: $\hat{M}+\hat{R}=180{}^\text{o}-\hat{A}=180{}^\text{o}-50{}^\text{o}=130{}^\text{o}$.

Num triângulo, a lados geometricamente iguais opõem-se ângulos geometricamente iguais. Ora, como $\overline{RA}=\overline{MA}$, então $\hat{M}=\hat{R}$.

Assim, $\hat{M}=\hat{R}=\frac{130{}^\text{o}}{2}=65{}^\text{o}$.

<< Enunciado