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MATH 11008:
Angles
Angle: An angle is the union of two line segments (or two rays) with a common endpoint,
called a vertex.
A
α
1
B
C
Adjacent angles: Adjacent angles are two angles that share a vertex, have a common
side, but whose interiors do not intersect.
A
B
C
D
Classification of Angles according to their measurements:
• Acute angle: angle measuring less than 90◦
• Right angle: angle measuring 90◦
• Obtuse angle: angle measuring more than 90◦ but less than 180◦
• Straight angle: angle measuring 180◦
• Reflex angle: angle measuring more than 180◦
2
MATH 11008: ANGLES
• Vertical Angles: Opposite angles formed by two intersecting lines are called vertical
angles.
IMPORTANT: Vertical angles always have the same measurement.
2
3
1
4
In the above figure, ∠1 and ∠3 are vertical angles; ∠2 and ∠4 are vertical angles.
• Complementary Angles: Two angles whose sum is 90◦ are called complementary
angles. If A and B are complementary angles, then A is the complement of B and
B is the complement of A.
1
2
In the above figure, ∠1 and ∠2 are complementary angles.
• Supplementary Angles: Two angles whose sum is 180◦ are called supplementary
angles. If A and B are supplementary angles, then A is the supplement of B and B
is the supplement of A.
1
2
In the above figure, ∠1 and ∠2 are supplementary angles.
MATH 11008: ANGLES
3
Example 1: Find the measure of each marked angle.
(a)
˚
˚
(11x - 34)
(4x +19)
(b)
(3x + 25)
(6x - 17)
(3x
-10
)
(c)
(6x+1)
4
MATH 11008: ANGLES
ANGLES ASSOCIATED WITH PARALLEL LINES:
• Corresponding Angles have the same location relative to lines `, m and transversal
t. (IMPORTANT: ` k m if and only if corresponding angles formed by `, m, and t are
congruent.)
In Figure A-1, ∠1 and ∠5 are corresponding angles. The following pairs are also corresponding angles: ∠2 and ∠6; ∠3 and ∠7; ∠4 and ∠8.
t
1
3
2
4
l
5
m
7
6
8
Figure A-1
• Alternate Interior Angles are nonadjacent angles formed by lines `, m, and transversal t, the union of whose interiors contain the region between ` and m. (IMPORTANT:
` k m if and only if alternate interior angles formed by `, m and t are congruent.)
In Figure A-1, ∠3 and ∠6 are alternate interior angles. Likewise, ∠4 and ∠5 are also
alternate interior angles.
• Alternate Exterior Angles are angles on the outer sides of two lines cut by a transversal, but on opposite sides of the transversal (IMPORTANT: ` k m if and only if alternate exterior angles formed by `, m and t are congruent.)
In Figure A-1, ∠2 and ∠7 are alternate exterior angles. Similarly, ∠1 and ∠8 are alternate exterior angles.
• Interior Angles on the same side of the transversal are interior angles whose
interiors are the same. (IMPORTANT: ` k m if and only if the interior angles on the
same side of the transversal are supplementary.)
In Figure A-1, ∠3 and ∠5, as well as ∠4 and ∠6, are interior angles on the same side of
the transversal.
MATH 11008: ANGLES
5
Example 2: In the diagram below, l k m and r k s. Find the measurement of each numbered
angle.
1
l
120
2
3
37
4
5
6
m
7
8
r
s
NOTES:
• The sum of the angles inside a triangle is 180◦ .
• The sum of the angles inside a quadrilateral is 360◦ .
Example 3: In the diagram below, BG k EF . Find the measure of each angle.
B
E
C
80˚
A
150˚
H
G
F
6
MATH 11008: ANGLES
Exercises
In #1–#6 find the measure of each marked angle.
1.
4.
(11x - 37)
(5x + 3)
(7x+27)
(4x + 6)
5.
2.
(x+1)
(11x - 4)
(4x - 56)
(8x + 17)
3.
(5x + 11)
6.
(3x - 15)
(3x-1)
(4x+7)
MATH 11008: ANGLES
7
7. Using the diagram below, name the relationship between the following pairs of angles,
given m k l and n k p.
4 3
12
1
2
5 6
7
11
10 9
15 16
13 14
m
n
8
p
l
a) ∠1 and ∠15
d) ∠11 and ∠10
g) ∠3 and ∠7
b) ∠1 and ∠5
e) ∠12 and ∠6
h) ∠14 and ∠10
c) ∠1 and ∠9
f) ∠6 and ∠8
i) ∠7 and ∠10
8. In the diagram below, m k n. Find the measure of each numbered angle.
1
40
3
122
4
2
5
m
7
6
8
n
9. In the diagram below, t k s and m k n. Find the measurement of each numbered angle.
t
115
s
1
3
4
m
6
2
5
n
8
MATH 11008: ANGLES
10. In the diagram below, l k m and r k s. Find the measurement of each numbered angle.
1 2
3
10
9
l
36
8
5
4
7
6
m
105
t
s
r
11. In the following figure, m k n and r ⊥ s. Given the angle measures indicated on the
figure, find the measure of each lettered angle.
r
m
50
b
c
n
e
105
d
s
a
f
12. Find the measure of x.
42˚
125˚
x˚
13. In the following figure, AB k CD. Find the measure of x and y.
B
x˚
70˚
D
y˚
A
50˚
C
MATH 11008: ANGLES
9
14. In the figure below, m k n. Find the measure of each labeled angle.
n
e
d
m
c
f
127
108
b
a
61
15. Find the measure of a and b.
b
a
32
44
16. Find the of each labeled angle.
50
c
b
a
i
110
60
d
30
e
f
h
g
40
17. In the figure below, m(∠BF C) = 55◦ , m(∠AF D) = 150◦ , and m(∠BF E) = 120◦ .
Determine the measures of ∠AF B and ∠CF D.
C
B
A
D
F
E
10
MATH 11008: ANGLES
ANSWERS
1. Both angles are 139◦
2. 48◦ , 132◦
3. 54◦ , 126◦
4. 48◦ , 42◦
5. Both angles are 73◦
6. 35◦ , 55◦
7. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
alternate exterior angles
alternate exterior angles
corresponding angles
interior angles on the same side of the transversal (supplementary angles)
alternate interior angles
corresponding angles
alternate interior angles
alternate interior angles
interior angles on the same side of the transversal (supplementary angles)
8. m(∠1) = 82◦ , m(∠2) = 58◦ , m(∠3) = 58◦ , m(∠4) = 82◦ , m(∠5) = 40◦ ,
m(∠6) = 58◦ , m(∠7) = 40◦ , m(∠8) = 140◦
9. m(∠1) = 65◦ , m(∠2) = 115◦ , m(∠3) = 65◦ , m(∠4) = 115◦ , m(∠5) = 115◦ ,
m(∠6) = 65◦
10. m(∠1) = 105◦ , m(∠2) = 75◦ , m(∠3) = 105◦ , m(∠4) = 105◦ , m(∠5) = 39◦ ,
m(∠6) = 36◦ , m(∠7) = 105◦ , m(∠8) = 39◦ , m(∠9) = 36◦ , m(∠10) = 105◦
11. m(∠a) = 40◦ , m(∠b) = 75◦ , m(∠c) = 55◦ , m(∠d) = 145◦ , m(∠e) = 35◦ ,
m(∠f ) = 125◦
12. m(∠x) = 83◦
13. m(∠x) = 50◦ , m(∠y) = 60◦
14. m(∠a) = 8◦ , m(∠b) = 64◦ , m(∠c) = 108◦ , m(∠d) = 72◦ , m(∠e) = 55◦ ,
m(∠f ) = 53◦
15. m(∠a) = 58◦ , m(∠b) = 104◦
16. m(∠a) = 70◦ , m(∠b) = 130◦ , m(∠c) = 100◦ , m(∠d) = 120◦ , m(∠e) = 20◦ ,
m(∠f ) = 20◦ , m(∠g) = 80◦ , m(∠h) = 60◦ , m(∠i) = 70◦
17. m(∠AF B) = 60◦ , m(∠CF D) = 35◦