Chapter 1 Lines and angles.pdf
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Chapter 1.qxd
3/8/09
6:50 PM
Page 1
CHAPTER
1 1.1
Lines and Angles II
Transversals and Parallel Lines
e E x a m p l e
To associate
CCTS
1 Find the value of x and of y of y in in the diagrams below below.. (a) y
x + 70 70°° = 180 180°° (an (angle gless on a str straig aight ht lin line) e) x = 110°
70°
x
(b)
y = y = 110° (cor (corres respon pondin ding g ang angles) les)
y
(c)
x
60°
x
55° 55 °
y
y
x + 60° 60° = 180° 180° (angles on a straight line) x = 120° (corresp respond onding ing y = 60° (cor angles)
x + 55° = 180° (angles on a straight line) x = 125° y = 55° (corresponding angles)
x
105°
x
+ 105 105°° = 180° 180° (ang (angle less on a straight line) x = 75° (correspondi ponding ng y = 75° (corres angles)
(d)
115° 115 °
x
y
115° = 180° 180° (angles on a x + 115° straight line) x = 65° (corresp respondi onding ng y = 65° (cor angles)
2 Find the value of x and of y of y in in the diagrams di agrams below.
e E x a m p l e
(a) 65°° x = 65
65°
y
x + y y = = 180° 180° (angle (angless on a straigh straightt line) y = 180° 65°° + y = 65 y = y = 115°
x
(b)
(c) x 100° 100 °
y
x + 100° 100° = 180° 180° (an (angle gless on a straight line) x = 80° = 80 80°° (a (alt lter erna nate te y = y angles)
x y
(alt (a lter erna nate te ang angle les) s)
y
110°
110° = 180° 180° (an (angle gless on a x + 110° straight line) x = 70° = 110° (altern (alternate ate y = y angles) (d) y
x 85°
85° = 180° 180° (an (angle gless on a x + 85° straight line) x = 95° = 85° 85° (a (alte ltern rnat atee y = y angles) 1
x
105°
105° = 180° 180° (angles on a x + 105° straight line) x = 75° 105° 5° (alternate y = 10 angles)) angles
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3 Find the value of m and of
n
in the following diagrams.
E x a m p l e
(a) m = 110° (alternate angles) m + n = 180° (interior angles) 110° + n = 180° n = 70°
m 110°
n
m
125°
n
m + 125° = 180° (interior angles) m = 55° n = 55° (corresponding angles)
(b)
(c)
(d) m
75°
n
m
m n
m + 75° = 180° (interior angles) m = 105°
115°
m = 115°(corresponding angles) m + n = 180° (interior angles) 115° + n = 180° n = 65°
n = 75° (alternate angles)
85°
n
85° + m = 180° (interior angles) m = 95° n
= 95° (alternate angles)
4 Find the value of x in the following diagrams.
E x a m p l e z = 65° (alternate angles) y + 110° = 180° (interior angles) y = 70°
65°
x
z
x
y 110°
(b) 3x
105°
x = 120° + 105° = 225°
105° 110°
120°
x
= y + z = 70° + 65° = 135°
(c) 2x
(a)
x
(d) 45°
x
55°
80°
3x + 2x + 80° = 180° (angles on a straight line) 5x = 100° x = 20°
70° + x + 75° = 180° (angles on a straight line) x + 145° = 180° x = 35° 2
x = 45° + 55° (alternate angles) = 100°
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PMR Practice 1 Paper 1 Answer all questions. 1 Which of the following diagrams shows line PQ parallel to line RS? 2008 A C P R P
4
A
G
B
110°
R
70°
y
°
C
D
E
70°
Q
50°
70°
70°
Q
S
B
P
P
R
In Diagram 3, CDE, ADF and GDH are straight lines. Find the value of y . A 60 C 70 B 65 D 80
70° 70° 70°
Q
H
DIAGRAM 3
D
R
F
S
S
Q
S
70°
5
M
2
L 120°
z
°
Q y
°
R
x
°
x
°
110°
50°
P
N
65°
J
K
S
DIAGRAM 4
In Diagram 4, JK is parallel to MN . Find the value of x. A 175 C 215 D 225 B 185
DIAGRAM 1
In Diagram 1, PQRS is a straight line. Which of the following is true? A x = y B x =z C y = z D x + y + z = 180
6
P
30°
x
°
R
45°
S
3
Q
T
Q
DIAGRAM 5 40°
P
In Diagram 5, PRT and QRS are straight lines. Find the value of x. A 75 C 95 B 85 D 105
S
R 60°
x
°
T
U
7
J
DIAGRAM 2
L
R T
S
80°
100°
In Diagram 2, PRS and QRT are straight lines. Find the value of x. A 35 B 40 C 50 D 60
P 55°
A B
C
D
125° 80°
K
M
DIAGRAM 6
3
Q
U
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In Diagram 6, JK , LM , PQ , RS and TU are straight lines. Some of the angles are given. Which of the points A, B, C or D, is situated between two parallel lines?
12
M
J
60°
2006
35°
L x
°
8 K
N
y
DIAGRAM 11
°
50°
P
70°
Q
R
S
In Diagram 11, JLN and KLM are straight lines. Find the value of x. C 95 A 85 B 90 D 105
DIAGRAM 7
In Diagram 7, PQRS is a straight line. Find the value of y . C 60 A 50 B 55 D 70 9
J
M
13
B
A
y
°
N
135°
125°
50°
D
E
F
x
°
K
L
In Diagram 12, ABC and DEFG are straight lines. Find the value of y . A 50 C 60 B 55 D 70
In Diagram 8, JMN is parallel to KL. Find the value of x. A 65 C 80 B 70 D 85 A
14 II
I
F
°
P
Q
K
S
R
S
L 15°
A D
Q
P
B
°
M
R
15 J
x
30°
In Diagram 13, PQRS is a straight line. Which of the following lines are parallel lines? A I and IV C II and V B II and IV D III and V
In Diagram 9, BCD and DEF are straight lines. Given that DC = DE, find the value of x. A 60 C 70 B 65 D 80 J
VI
DIAGRAM 13 D
DIAGRAM 9
11
45°
30°
C
V
30° 20°
65°
x
IV
III
35°
E
B
G
DIAGRAM 12
DIAGRAM 8
10
C 75°
120°
K
C N
30° 20°
45°
L
M
DIAGRAM 14
P
In Diagram 14, KLM is a straight line. Which of the following lines is parallel to the line JK ? A PL C RL B QL D SL
DIAGRAM 10
In Diagram 10, JKL and MNP are straight lines. Which of the angles A, B, C or D, is equal to x?
4
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16
20
K J
L
115°
T 50° 85°
x
°
P
y
°
R
Q
S
DIAGRAM 15
DIAGRAM 19
In Diagram 15, PQR and SQT are straight lines. Find the value of x. A 55 C 40 D 35 B 45
In Diagram 19, JKL is a straight line. Find the value of y . A 180 C 195 D 205 B 190
17
21 100°
P
2004
J
135°
y
A
R
K
B
°
P
Q
L
R
D
C
Q
18
DIAGRAM 20
In Diagram 20, JK, LM, PQ and RS are straight lines. Which of the angles A, B, C or D, is equal to x?
J
22 K
2x ° 3x °
L
45°
130°
J M
125°
K
L
°
In Diagram 21, JKL is a straight line. Find the value of x. A 55 C 75 D 80 B 70
In Diagram 17, JKLM is a straight line. Find the value of x. A 10 C 20 B 15 D 26 13 cm
P
x
DIAGRAM 21
DIAGRAM 17
19
°
S
DIAGRAM 16
In Diagram 16, PQR is a straight line. Find the value of y . C 55 A 45 B 50 D 65
23
Q
25°
y
°
5 c m
R
M
x
1 2
x
°
c m
x
°
P 115°
S
Q
R
DIAGRAM 18
DIAGRAM 22
In Diagram 18, PQ and RS are straight lines. Find the value of x + y . C 180 A 90 B 135 D 270
In Diagram 22, PQR is a straight line. Find the value of x. A 90 C 130 D 140 B 125
5
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24
28
J
T
P
K x
°
240° 110°
Q
x
°
R
125°
S
L
M
DIAGRAM 23
DIAGRAM 27
In Diagram 23, QRS is a straight line and QP = QR. Find the value of x. C 140 A 70 B 130 D 150
In Diagram 27, JK is parallel to LM . Find the value of x. C 115 A 95 B 110 D 120
25
29
S
85°
x
°
y
35°
°
P
P
R
Q
DIAGRAM 28
DIAGRAM 24
In Diagram 28, PQ is parallel to ST . Find the value of x. A 135 C 150 B 145 D 205
In Diagram 24, PQR is a straight line. Find the value of y . A 35 C 55 B 50 D 60
30
J
C °
°
B
s
°
p
°
A
r
q
K
F 235°
x
°
M
N
DIAGRAM 29
In Diagram 25, ABC and DEF are straight lines. Which of the following is true? A p = r C q = r B p=s D q + s = 180
In Diagram 29, JKL is a straight line. Find the value of x. A 105 C 120 D 130 B 115
P
31
70°
2006
K x
M J
y
°
56° 40°
°
L
P
L
DIAGRAM 25
J
75°
E
D
27
R
125°
Q
26
T
80°
x
°
L
M K
N
Q
DIAGRAM 30
DIAGRAM 26
In Diagram 30, JLN and KLM are straight lines. Find the value of x. C 96 A 80 B 84 D 102
In Diagram 26, JK, LM and PQ are straight lines. Find the value of x – y . A 30 C 50 B 40 D 55
6
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32
34 A
C
E
x
°
65° 70° 60°
70° 50°
60°
B
y
°
50°
D
20°
J
F
K
L
DIAGRAM 33 DIAGRAM 31
In Diagram 31, BDF is a of the following pairs of A AB and CD C B AB and EF D 33
In Diagram 33, JKL is a straight line. Find the value of x + y . A 100 C 120 B 110 D 135
straight line. Which lines is parallel? BC and DE CD and EF
35 K
J
L
R
M
P
S
2005
a b
K
20°
25°
20°
P
L
c
R
110°
30°
y
e
M
f
Q
T
DIAGRAM 32
d
U N
x Q
DIAGRAM 34
In Diagram 32, JP , KP , LP , MP , PQ and QR are straight lines. Which of the following lines is parallel to QR? A JP C LP B KP D MP
In Diagram 34, KL, MN , PQ , RS and TU are straight lines. The value of x + y is the same as C b+e A a + d B a + f D c + f
Paper 2 Answer all questions. 1 In Diagram 1, PQRS is a straight line. Find the value of x.
2 In Diagram 2, PQRS is a straight line. Find the value of x.
P 130°
Q x
°
x R °
P S
Q
y
°
R
70°
S
DIAGRAM 2
DIAGRAM 1
70° + 70° + y = 180° (sum of angles in ∆) y = 40° x + y = 180° (angles on a straight line) x = 140°
x + 130° = 180° (angles on a straight line) x = 50°
140° Answer: ______________
50° Answer: ______________
7
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3 In Diagram 3, JK is parallel to NM . Find the value of x. J
4 In Diagram 4, DEF is a straight line. Find the value of x.
2003
N
105°
x K °
°
°
D
L
P
135°
x
°
50° Answer: ______________
In Diagram 5, PQ is parallel to RS. Find the value of x + y .
R
6 In Diagram 6, JKL and LMN are straight lines. Find the value of y . J
N
50°
S y
°
K
M
50° + y + 85° = 180° (angles on a straight line) y + 135° = 180° y = 45°
y
°
DIAGRAM 5
85°
L
45° + x + y = 360° (angle of one whole turn) x + y = 360° – 45° = 315°
DIAGRAM 6
45° Answer: ______________
315° Answer: ______________ In Diagram 7, DEF and EGH are straight lines. Find the value of x.
7 H x
°
G
E
120°
x
°
110°
F
DIAGRAM 7
DIAGRAM 8
x + 60° + 70° = 360° (angle of one whole turn) x + 130° = 360° x = 230°
x + 90° + 60° = 180° (interior angles) x + 150° = 180° x = 30° 30° Answer: ______________ 9 y
°
L
230° Answer: ______________
In Diagram 9, JK is parallel to LM . Find M the value of y .
In Diagram 10, JK is parallel to M MN . Find the x value of x.
10 J
°
120°
K
125°
J
Diagram 8 shows two parallel lines. Find the value of x.
8
50°
70°
D
F
65° + 65° + y = 180° (sum of angles in ∆) 130° + y = 180° y = 50° = y = 50° (corresponding angles) x
60° + 75° + x = 360° (angle of one whole turn) x + 135° = 360° x = 225° 225° Answer: ______________
Q
65°
E
DIAGRAM 4
DIAGRAM 3
5
y
x
M
60°
K
L 235°
DIAGRAM 9
N
DIAGRAM 10
y + 90° + 125° = 360° (angle of one whole turn) y + 215° = 360° y = 145° 145° Answer: ______________
60° + x + 235° = 360° (angle of one whole turn) x + 295° = 360° 65° x = 65° Answer: ______________
8
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11 In Diagram 11, PQ is parallel to ST . Find the value of x. P
12 In Diagram 12, JK is parallel to MN . Find the value of x. M
Q
N 50°
105°
S
T 120°
x
°
4x ° + 15°
L
115°
R J
K
DIAGRAM 11 DIAGRAM 12
60° + x + 75° = 180° (sum of angles in x + 135° = 180° x = 45°
∆)
4x + 15° = 50° + 65° 4x = 115° – 15° 4x = 100° x = 25°
45° Answer: _____________________ 13
In Diagram 13, RSTU is a straight line. Find the value of x and of y.
R y
°
2x °
T
25° Answer: ______________
S
14 In Diagram 14, AB, CD and EF are straight lines. Find the value of x. E
4x ° – 60°
A
U
70°
B 2x ° + 30°
C
DIAGRAM 13
F
2x = 4x – 60° (alternate angles) 2x = 60° x = 30° y = 2x (vertically opposite angles) = 2 × 30° = 60°
DIAGRAM 14
70° + 2x + 30° = 180° (angles on a straight line) 2x = 180° – 100° x = 40°
x = 30°, y = 60° Answer: _____________________
40° Answer: ______________
15 In Diagram 15, PQ, RS and TU are straight lines. Find the value of y .
16 In Diagram 16, PST is parallel to RU . PQR is a straight line. It is given that PS = PQ . Find the value of x.
T
T P
Q
65°
x
°
R
°
U
S
S
y + 15
°
y
°
P
U
DIAGRAM 15
x = x + y + 15° = 65° + y + 15° = y =
D
125°
x
Q
°
R
DIAGRAM 16
65° (alternate angles) 180° (angles on a straight line) 180° 100°
y + 55° + 55° = 180° (sum of angles in ∆) y = 70° x + y = 180° (interior angles) x = 110° 110° Answer: ______________
100° Answer: _____________________
9
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