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Slide 1 PARALELISMO r s t α r C α : r está contido no plano α s C α t C α Retas coplanares ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 2 r s α r // s Retas paralelas ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 3 Retas concorrentes: r e s – possuem um único ponto em comum r s t α P D ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 4 Ângulos formados por duas retas concorrentes A B C D O AÔD AÔB DÔC CÔB Quais os ângulos congruentes? Caso especial: ângulos retos! ≅ ≅ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 5 Ângulos formados por duas retas coplanares e uma transversal s t r Região externa Região interna Região externa ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 6 s t r Região externa Região interna Região externa A transversal t divide o plano em duas regiões, determinando ângulos numa mesma região ou em regiões alternadas em relação a esta transversal ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 7 Ângulos correspondentes Mesmo lado da transversal, sendo um interno e outro externo ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 8 Ângulos alternos internos Estão na região interna das coplanares r e s, alternados em relação a t ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 9 Ângulos alternos externos Estão na região externa das coplanares r e s, alternados em relação a t ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 10 Ângulos colaterais internos (e externos) Estão na região interna (externa) das coplanares r e s, situados do mesmo lado da transversal t ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 11 r s r//s O P C B A D E G F Exercício 1 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 12 r s r//s ! 960 510 48O Exercício 2 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 13 Ângulos e Diagonais de um Polígono Ângulos internos de um triângulo a b c a b c r s a+b+c=180o ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 14 Ângulos externos de um triângulo a b c r se e=a+b ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 15 Na figura seguinte, sendo a reta r paralela à reta s, a sentença verdadeira é: r s t 42o x y z a) y mede 138o e x e z são alternos internos b) x mede 42o e x e z são correspondentes c) y mede 42o e x e z são alternos internos d) x mede 138o e x e z são opostos pelo vértice. Exercício 3 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 16 Na figura a seguir, tem-se r//s e t e u transversais. Qual o valor de α + β? u r s t 20o α 70o β Exercício 4 ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 17 Sabendo-se que t//u, qual o valor de y? t u 100o y 60o Exercício 5 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 18 Qual a medida do ângulo x, sabendo que r//s? r s 120o x 160o Exercício 6 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 19 Na figura a seguir, tem-se BC ≅ CD e AB // CE. Qual o valor de x? A D E B C 32o 126o x Exercício 7 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 20 Quando duas retas paralelas coplanares r e s são cortadas por uma transversal t, elas formam: a) ângulos alternos externos suplementares b) ângulos colaterais internos complementares c) ângulos alternos externos e internos congruentes d) ângulos correspondentes congruentes Exercício 8 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 21 Ângulos internos de um polígono B F E D C A Teremos tantos triângulos quantos forem os lados menos 2 6 lados (6 – 2) triângulos ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 22 Quadriláteros = 4 lados Pentágono = 5 lados Decágono = 10 lados Hexágono = 6 lados Octógono = 8 lados Heptágono = 7 lados Pentadecágono = 15 lados Icoságono = 20 lados Eneágono = 9 lados ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 23 De um modo geral, se um polígono tiver n lados, teremos: n lados (n – 2) 180o A soma dos n ângulos internos de um polígono convexo de n lados é: n lados (n – 2) triângulos Polígono Regular: In = (n – 2) 180 o n ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 24 Diagonais de um Polígono B F E D C A Dn= n . (n-3) 2 Número de diagonais de um polígono de n lados ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 25 Ângulos externos de um polígono B F E D C A a’ f’ d’ e’ c’ b’ a f e d c b a+a’=180o b+b’=180o c+c’=180o d+d’ =180o e+e’=180o f+f’=180o ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 26 Quadriláteros Polígonos de quatro lados paralelogramo trapézio Tem os lados opostos paralelos Tem apenas dois lados opostos paralelos ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 27 Classificação dos paralelogramos -losangos -retângulos -quadrados Propriedades: *Em todo paralelogramo, os lados opostos são congruentes *Em todo paralelogramo os ângulos opostos são congruentes *Em todo paralelogramo, as diagonais se cortam mutuamente ao meio ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 28 Propriedades dos losangos Em todo losango as diagonais são perpendiculares Em todo losango as diagonais são bissetrizes dos ângulos internos Propriedades dos retângulos Em todo retângulo as medidas das diagonais são iguais ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 29 CLASSIFICAÇÃO DOS TRAPÉZIOS Trapézio retângulo Trapézio isósceles lados não-paralelos congruentes Propriedades dos trapézios Em todo trapézio isósceles, os ângulos adjacentes a uma mesma base são congruentes Em todo trapézio, a base média é paralela às outras base e mede a semi-soma das medidas dessas bases ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 30 A Circunferência e o Círculo Dados um ponto O de um plano e uma distância r, chama-se circunferência de centro O e raio r ao conjunto dos pontos que distam r de O. r O Ao conjunto de pontos obtido pela reunião da circunferência com a região interna a ela chamamos de círculo r O ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 31 Elementos da circunferência A D B C Corda da circunferência: AB es CD Diâmetro da circunferência: CD E P F Arcos da circunferência: EF e EPF ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 32 POSIÇÕES RELATIVAS t é externa a C r é tangente a C s é secante a C t rs ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 33 Posições relativas: duas circuferências C D C’’ C’ C’’’ P E F C’’’ é externa a C C e C’ são tangentes externamente C e C’’’’ são tangentes internamente C’’’’ C e C’’ são secantes ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 34 Ângulos e arcos de uma circunferência B A O Ângulo central B A O Ângulo inscrito O’ med(AÔB) = med(AB) ! med(AÔ’B) = med(AB) ! 2 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 35 AO O’ B AÔ’B é um ângulo de segmento Ângulo de segmento é o ângulo inscrito onde um dos lados é uma reta tangente à circunferência ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 36 Ângulo excêntrico interior : tem o vértice interior à circunferência e não-coincidente com o centro Ângulo excêntrico exterior : tem o vértice exterior à circunferência e os lados são semi-retas que possuem pelo menos um ponto comum com essa circunferência O E A B D med(AÊB) = med(AB) + med (CD)"""" """" 2 med(AÊB) = med(AB) - med (CD)"""" """" 2 C C O A B D ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 37 EXERCÍCIOS 9. Determine o valor de x na figura 30o 20o x ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 38 10. Cálcule a medida do ângulo interno dos seguintes polígonos regulares: Quadriláteros = 4 lados Pentágono = 5 lados Decágono = 10 lados Hexágono = 6 lados Octógono = 8 lados Heptágono = 7 lados Pentadecágono = 15 lados Icoságono = 20 lados Eneágono = 9 lados ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 39 11. Num polígono regular, o ângulo interno é o dobro do ângulo externo. Quantos lados tem esse polígono? 12. Num polígono regular, o ângulo interno é igual ao ângulo externo. Quantos lados tem esse polígono, e quanto mede cada ângulo? 13. Num polígono regular, cada ângulo interno mede 135o. Quantas diagonais tem esse polígono? 14. Num polígono convexo, a soma das medidas dos ângulos internos excede à soma das medidas dos ângulos externos em 540o. Qual é esse polígono? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 40 15. Qual é o polígono regular cuja medida do ângulo interno é o quíntuplo da medida do ângulo externo? 16. Num quadrilátero, a medida de cada ângulo supera a medida do anterior em 40o. Calcule as medidas dos ângulos desse quadrilátero. 17. O arco AB é a metade do arco CD. Traçam-se DA e CB, que se cortam num ponto exterior, formando um ângulo de 35o. Determine as medidas dos arcos AB e CD. " " "" ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 41 18. São dadas duas circunferências secantes, de centro O1 e O2, cujos raios medem, respectivamente, 9 cm e 17 cm. Sendo x a distância entre os centros O1 e O2, o que poderemos concluir sobre essa distância? O1 O2x 17 9 |r-r’|<x<|r+r’| 8<x<26 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 42 REVISÃO 1. Dados dois polígonos regulares, com (n+1) lados e n lados, respectivamente. Determine n, sabendo que o ângulo interno do primeiro polígono excede o ângulo interno do segundo em 5 graus. 2. São dadas duas circunferências secantes, de centros O1 e O2, cujos raios medem, respectivamente, 5 m e 10 m. Sabendo-se que a distância entre os pontos de intersecção das circunferências é de 8 m, determine a distância entre os centros O1 e O2. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 43 3. Entre duas retas paralelas r e s tem-se o ponto de intersecção de duas retas concorrentes t e m. A transversal t faz um ângulo de 30o com a reta r e a transversal m faz um ângulo de 45o com a reta s. Determine os ângulos internos dos triângulos formados pelas retas paralelas e as retas transversais. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 44 4. As circunferências são tangentes em Q e PA e PB são tangentes às circunferências. Determine a medida do ângulo AQB nos casos: a) onde t é tangente comum e APB = 80 graus P B Q A t ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 45 b) com APB = 100 graus P Q A B ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________ Slide 46 5) O arco AB é a metade do arco CD. Traçam-se DA e CB, que se cortam num ponto exterior, formando um ângulo de 35o. Determine as medidas dos arcos AB e CD. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ____________________________________________________________________
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