Download 6.2 Geometry Problems Angles with Parallel Lines

6.2 Geometry Problems
Angles with Parallel Lines
- corresponding angles are
equal b = f, c = g
a
- alternate interior angles are
equal c = f, d = e
c
g
d
f
e
- alternate exterior angles are
equal b = g, a = h
b
h
- a and b are supplementary
Eg)
find angles a and b
b
o
37
a
o
A = 90 - 37 = 53
B = A = 53o
Eg)
find angle a
40
A = 180 - 40 = 140
A
Homework pg 165 1 – 21 do odd
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Eg)
When two straight lines intersect, the opposing angles are equal.
If M = 30D , find x
180 = 30D + X
M
X
M
X
X = 150D
When parallel lines are cut by a third line,
1) the corresponding angles are equal
X or M – corresponding angles
X
X
2) the alternate interior angles are equal.
O – alternate interior angles
O
O
Find the angles
a) 1) 180 − 40 = 140D
b) 5) 40D - corresponding
c) 2) 40D - alternate interior
d) 4) 140D
1
6
Find the angles
a) 3) 130D
b) 2) 130D
c) 4) 180 − 130 = 50D
d) 5) 50D
1
2
5
4
O
3
7
2
3 4
5
130
7
2
O
8
Homework pg 166 23 – 33 do odd
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Polygons
- a polygon is a closed figure whose sides are straight lines
triangle
Quadrilateral
pentagon
hexagon
septagon
octagon
nomagon
decagon
Number of sides
3
4
5
6
7
8
9
10
Parallelogram - parallel opposite sides
Eg)
Rectangle
Area = length × width
Rhombus
Area = base % height
Trapezoid
Area =
a+b
2 h
height or
altitude
height or
altitude
Find the cost to pave a parking lot which measures 80 m by 60 m. The cost to pave 1 m2 is
$15.30.
A = l × w = 80 × 60 = 4800 m 2
Cost = 4800 × 15.30 = $73, 440
The cost to pave the parking lot is $73,440
Eg)
The dimensions of the trapezoidal face of a bridge abutment have parallel sides of 10 m and 12
m with a height of 7 m. Find the area.
A = a + bh
2
= 10 + 12 % 7 = 77m 2
2
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10 m
12 m
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Eg)
Find the height of a flowerbed in the shape of a rhombus. The base length is 14 m and its area
is 140 m2.
A = base % height
140 = 14h
height = 140 = 10m
14
Eg)
14 m
How many lots are created when a 9,600 m2 rectangular plot of land is subdivided into lots 20
m by 40 m?
Lot area = 20 X 40 = 800 m2
The number of lots = 9600 / 800 = 12 lots
Eg)
A yard which is 40ft by 60 ft is to be re-soded with sod rolls which are 3 ×1
1
ft . How many rolls
2
should be ordered?
Yard area= 40 × 60 = 2400 ft 2
Roll area= 3 × 1 1 = 4.5 ft 2
2
2400
= 533.3
# of rolls=
4.5
Eg)
Find the area.
Base= 3.2 + 7.2 = 10.4
Height = 5.522 = 3.22 + h 2
h 2 = 5.522 − 3.22
h = 4.47
1
1
A = bh = 10.4 × 4.47 = 23.2
2
2
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5.52
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8.49
7.20
Eg)
A metal plate which is 4.9 inches by 3.6 inches contains a circular hole with a radius of 1.6
inches and a square hole with a side length of 1.5 inches. Calculate the weight of the plate
given that 1in2 weighs 2.5 oz.
Area = ( 4.9 × 3.6 ) − π (1.6 ) − 1.5 × 1.5
2
= 17.64 − 8.04 − 2.25
= 7.35in 2
Weight = 7.35 × 2.5 = 18.38
Eg)
Find the perimeter and area.
45ft
c 2 = 402 + 452 = 3625
40ft
c = 60.21
c 2 = 1002 + 902 = 10000 + 8100
100ft
c = 18100
2
c = 134.54
Perimeter = 45 + 60.21 + 134.54 + 140
= 379.8 ft
90ft
⎛1
⎞ ⎛1
⎞
Area = ( 45 × 40 ) + ⎜ 40 × 45 ⎟ + ⎜ 100 × 90 ⎟
⎠
⎝2
⎠ ⎝2
= 1800 + 900 + 4500
= 7200 ft 2
Eg)
Find the area
8”
A = ( 20 ×19 ) − ( 8 × 8 )
20”
12”
= 380 − 64 = 316in 2
19”
Eg)
Find the filled in area
4m
1
1
(8)( 7 ) − (8)( 3)
2
2
= 28 − 12 = 16m 2
A=
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8m
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Eg)
A sheet of metal that is 1.5m wide is bent into a circular tube. What is the radius of the tube?
Use π = 3.14159
c = 2π r
1.5 = 2π r
1.5
1.5
r=
=
= 0.239m
2π 6.283
Eg)
The window frame cost $4.50 per foot.
What is the total cost of the two pieces of the frame?
⎛8⎞
c = 2π r = 2π ⎜ ⎟ = 25.133
⎝2⎠
25.133
arc length =
= 12.57 ft
2
Length = 12.57 + 8 = 20.57 ft
Cost = 20.57 × 4.50 = $92.57
Eg)
8ft
Find the inner area of the ring given that the ring thickness is 2cm.
A = π (10 + 2 ) − π 102
2
10cm
= 452.39 − 314.16
= 138.23cm 2
Eg)
2cm
Find the area of the circle which contains a square with a side length of 5m.
Diameter 2 = 52 + 52 = 25 + 25 = 50
Diameter = 50 = 7.07 m
2
⎛ 7.07 ⎞
2
A = π r2 = π ⎜
⎟ = 39.26m
⎝ 2 ⎠
Eg)
If a medium pizza with a 10 inch diameter will serve 2 people, how many people should a large
pizza serve with 14 inch diameter?
A m = π r 2 = π 52 = 78.54in 2 → 2people
A l = π 7 2 = 153.93in 2 → xpeople
x
2
=
78.54 153.93
x=
2 ×153.93
= 3.91people
78.54
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Eg)
Two race horses are racing around a circular track. Horse A is in a track 450m from the center,
while horse B is running 456m from the center. Which horse has the advantage and by what
amount?
C A = 2π ( 450 ) = 2827.4m
CB = 2π ( 456 ) = 2865.1m
Horse A has an advantage of 2865.1 − 2827.4 = 37.7m
Eg)
Find the perimeter and area of an equilateral triangle with one side equal to 10 m.
Perimeter = 3 X 10 = 30 m
10
area =
1
2 bh
=
1
2
10
% 10 % 8.66 = 43.3m 2
height = 10 2 − 5 2 = 8.66
10
Similar Polygons
- two polygons are similar if all the corresponding angles are equal
B
B
C
C
D
D
E
E
A
A
A
- triangles ABC and ADE are similar since
y The corresponding angles are equal
y The corresponding sides are proportional
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Eg)
Find the lengths AN, CN and AM if ΔABC and ΔAMN are similar
A
5
C
AN = MN
AC BC
AN = 12
5
7
AN = 12 % 5 = 8.57
7
7
B
N
12
M
CN = AN - AC = 8.57 - 5 = 3.6
AM cannot be solved since there is insufficient information.
Eg)
Find length x
Since sides AB and CD are parallel lines with diagonals
Then angles C = B and D = A
B
10
A
x = 4
11 10
x = 4 % 11 = 4.4
10
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11
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D
4