apostila2_geometria

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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
 
 
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Slide 2 
 
r
s
α r // s
Retas paralelas
 
 
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Slide 3 
 
Retas concorrentes: r e s – possuem um único ponto em comum
r
s
t
α
P
D
 
 
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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!
≅
≅
 
 
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Slide 5 
 
Ângulos formados por duas retas coplanares e uma transversal
s
t
r
Região externa
Região interna
Região externa
 
 
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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
 
 
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Slide 7 
 
Ângulos correspondentes
Mesmo lado da transversal, 
sendo um interno e outro 
externo
 
 
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Slide 8 
 
Ângulos alternos internos
Estão na região interna das 
coplanares r e s, alternados 
em relação a t
 
 
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Slide 9 
 
Ângulos alternos externos
Estão na região externa das 
coplanares r e s, alternados 
em relação a t
 
 
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Slide 10 
 
Ângulos colaterais internos (e externos)
Estão na região interna 
(externa) das coplanares r e 
s, situados do mesmo lado 
da transversal t
 
 
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Slide 11 
 
r
s
r//s
O
P
C
B
A
D
E
G
F
Exercício 1
 
 
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Slide 12 
 
r
s
r//s
!
960
510
48O
Exercício 2
 
 
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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
 
 
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Slide 14 
 
Ângulos externos de um triângulo
a
b c
r
se
e=a+b
 
 
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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
 
 
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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
 
 
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Slide 17 
 
Sabendo-se que t//u, qual o valor de y?
t
u
100o
y
60o
Exercício 5
 
 
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Slide 18 
 
Qual a medida do ângulo x, sabendo que r//s? 
r
s
120o
x
160o
Exercício 6
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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Slide 26 
 
Quadriláteros
Polígonos de quatro lados
paralelogramo
trapézio
Tem os lados opostos paralelos
Tem apenas dois lados opostos paralelos
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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Slide 32 
 
POSIÇÕES RELATIVAS
t é externa a C
r é tangente a C
s é secante a C
t
rs
 
 
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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
 
 
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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
 
 
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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
 
 
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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
 
 
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Slide 37 
 
EXERCÍCIOS
9. Determine o valor de x na figura
30o
20o
x
 
 
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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
 
 
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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?
 
 
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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. 
" "
""
 
 
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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
 
 
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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.
 
 
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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.
 
 
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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
 
 
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Slide 45 
 
b) com APB = 100 graus 
P
Q
A
B
 
 
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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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