SECTION 4.6
4.6 ANGLE MEASUREMENT
CLAST OBJECTIVE
" Identify relationships between angle measures
Angle Measurement 169
Textbook Reference Section 8.1
Angle measurement is the measurement between two rays from a common point. Angles are
usually measured in degrees.
Name of Angle
Basic Shape
Angle Measurement
Acute
Less than 90 º
Obtuse
Greater than 90 º
Right
Exactly 90º
Straight
Exactly 180 º
FACTS
a) Complementary angles sum to 90 º.
b) Supplementary angles sum to 180 º.
c) Opposite angles formed by two intersecting lines are equal.
d) Two parallel lines cut by a transversal usually create four equal obtuse angles and four
equal acute angles.
obtuse
acute
obtuse
acute
acute
obtuse
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acute
obtuse
170 CHAPTER 4
Geometry
FACTS
e) The sum of the angles in a polygon is 180( n – 2 ), where n is the number of sides of
the polygon.
f) Parallel lines never intersect.
g) Perpendicular lines intersect at 90º .
Examples
a) What angle is complementary to 75 º ?
Solutions
75 º + x = 90 º
x = 90 º - 75 º Complementary
angles sum to 90º.
x = 15 º
b) What angle is supplementary to 60 º ?
60 º + x = 180 º
x = 180 º - 60 º Supplementary
angles sum to180º.
x = 120 º
c) Identify two pairs of opposite angles.
Angles 1 and 4
Angles 2 and 3
1
3
2
4
Note that the two angles together form a
straight angle. This means the angles are
supplementary and must sum to 180 º.
50 º + x = 180 º
x = 180 º - 50 º = 130 º
d) Find angle x.
x
50º
e) Give the measure of angles 1, 2, and 3.
1
2
4
6
3
5
7
60º
∠ 1 and 60 º are supplementary.
∠ 1 + 60 º = 180 º
∠ 1 = 180 º - 60 º
∠ 1 = 120 º
∠ 1 and ∠ 3 are opposite angles.
Thus, ∠ 1 = ∠ 3 . Angle 3 measures 120 º .
The 60 º angle and ∠ 2 are also opposite
angles. ∠ 2 = 60 º .
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SECTION 4.6
Angle Measurement 171
Examples
f) What is the measure of one angle of a
regular pentagon?
Solutions
A pentagon has five sides. Use the sum of
angles formula, 180( n – 2 ), where n = 5.
180 ( 5 – 2 ) = 180 (3) = 540 º
The sum of the five angles in a pentagon is
540 º. To find the measure of just one
angle, divide the total by 5.
540 °
= 108 °
5
Each angle in a regular pentagon measures
108 º.
g) If L 1 and L 2 are parallel and L 3 and L 4
are parallel, which of the following
statements are true?
x = 55 º because x and the 55 º angle are
corresponding angles.
L1
y = 180 º – 55 º = 125 º , since x and y
are supplementary.
L2
s
t
z
55º
L3
r
x
y
z = y = 125 º, since z and y are alternate
interior angles
L4
Remember, two parallel
lines cut by a transversal
usually create four equal
obtuse angles and four
equal acute angles.
a) z = 55 º
b) r = 125 º
c) z = 125 º
d) s = t
Check Your Progress 4.6
1. Name the type of angle.
a)
The correct answer is C.
b)
c)
2. Find the complement of the following:
a) 20 º
b) 50 º
c) 70 º
3. Find the supplement of the following:
a) 40 º
b) 100 º
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c) 135 º
172 CHAPTER 4
Geometry
4. Identify the pairs of opposite angles.
t
v
u
w
5. Find the angle measure of x.
a)
b)
140 º
80 º
x
x
6. Lines L 1 and L 2 are parallel. Find measurement of angles 1, 2, and 3.
L1
1
2
3
L2
30º
7. Find x if lines L 1 and L 2 are parallel.
L1
L2
3x
4x - 7
8. Find the measure of the angles if lines L 1 and L 2 are parallel.
L2
L1
3x + 10
2x
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SECTION 4.6
Angle Measurement 173
9. Find angle x.
30 º
x
10. Find the sum of the angles of a hexagon.
11. Find the sum of the angles of a rectangle.
12. Find the measure of an inside angle of a regular octagon.
13. Which statement is true for the figure shown below given that L 1 and L 2 are
parallel lines?
a) Since t = 80 º, s = 80 º
b) Since t = 80 º, s = 100 º
c) v = r
d) r = 55 º
q
L1
p
r
80 º
L2
u
s
v
t
w
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x
y
55 º
174 CHAPTER 4
Geometry
See If You
Remember
SECTIONS 4.1 – 4.5
A
1
2
1. Round position A to the nearest inch.
3
4
5
2. Round position A to the nearest ½ inch.
3. Jack wants to put wire around a square window of side 4.5 ft. How much wire must
he buy?
4. Ben is carpeting a floor 12 feet by 9 feet. How many square yards of carpet does he
need?
5. A milk can is 8 inches tall and 2 inches in diameter. What is the volume of milk in a
full can?
6. Find the surface area of a cube of side 4.3 ft.
For Questions 7 and 8, write an appropriate measure.
7. The distance around a pool
8. The amount of water in the pool
9. Find the area of the shaded region.
4 cm
4 cm
4 cm
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