Chapter 1 Lines and angles.pdf

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1.1

1.1 Transversals and Parallel Lines

Transversals and Parallel Lines

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1 Find the value ofFind the value of xx and ofand of y y in the diagrams in the diagrams belowbelow..

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2 Find the value ofFind the value of xx and ofand of y y in the diin the diagrams below.agrams below.

Lines and Angles II

CHAPTER

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E

x

a

m

p

l

e

(a) (a) x x + 1+ 10505°° = 18= 180°0° (a(angngleles on as on a straight line) straight line) x x= 75°= 75° y

y = = 75°75°(corres(correspondipondingng

angles) angles) 105° 105° x x y y (d) (d) x x + 11+ 115°5°= 18= 180°0° (angles on a(angles on a straight line) straight line) x x = 65°= 65° y

y = 65°= 65° (cor(corresprespondiondingng

angles) angles) 115 115°° _x_x y y (c) (c) x x + 55°+ 55° = 180°= 180° (angles on a(angles on a straight line) straight line) x x = 125°= 125° y

y = 55°= 55° (corresponding(corresponding

angles) angles) x x 5555°° y y (b) (b) x x + 60+ 60°°= 18= 180°0°(angles on a(angles on a straight line) straight line) x x = 120°= 120° y

y = 60°= 60°(cor(corresprespondondinging

angles) angles) 60 60°° x x y y x

x + 70+ 70°°== 180180°°(an(anglegles on a ss on a strtraigaight ht linline)e) x

x == 111100°° y

y == 110°110°(cor(corresresponpondinding g angangles)les)

To associate To associate

CCTS

E

x

a

m

p

l

e

(a) (a)

x

x + 11+ 110°0°= 18= 180°0°(an(anglegles on s on aa

straight line)

x

x = 70°= 70°

y

y = = 110°110°(altern(alternateate

angles) angles) x x ₁₁₀₁₁₀ ° ° y y (d) (d) x x + 10+ 105°5°= 18= 180°0°(angles on a(angles on a straight line) straight line) x x = 75°= 75° y

y = = 10105°5° ((alternatealternate

angles angles)) 105 105°° x x y y (c) (c) x

x + 85+ 85°° = 18= 180°0°(an(anglegles on s on aa

straight line)

x

x = 95°= 95°

y

y = 85= 85°° (a(altelternrnatatee

angles) angles) 85 85°° x x y y (b) (b) x

x + 10+ 100°0°= 18= 180°0°(an(anglegles on s on aa

straight line) straight line) x x = 80°= 80° y y = = 8080°° (a(altlterernanatete angles) angles) 100 100°° x x y y x = x = 6565°° (a(altlterernanate ate angngleles)s) x

x ++ y y = 180= 180°° (angle(angles on a sts on a straighraight line)t line) 65 65°° ++ y y = 180°= 180° y y = 115°= 115° 70 70°° y y x x 65 65°° x x y y

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4 Find the value of x in the following diagrams.

E x a m p l e

(a) m + 125° = 180° (interior angles) m = 55° n = 55° (corresponding angles) 125° n m (d) 85° + m = 180° (interior angles) m= 95° n= 95° (alternate angles) 85° m n (c) m = 115°(corresponding angles) m + n = 180° (interior angles) 115° + n= 180° n= 65° m n ₁₁₅ ° (b) m + 75° = 180° (interior angles) m = 105° n = 75° (alternate angles) 75° m n m = 110° (alternate angles) m + n = 180° (interior angles) 110° + n= 180° n= 70°

E x a m p l e

(a) x = 120° + 105° = 225° 120° 105° x (d) x = 45° + 55° (alternate angles) = 100° 45° _x 55° (c) 70° + x + 75° = 180° (angles on a straight line) x + 145° = 180° x = 35° 105° 110° _x (b) 3x + 2x + 80° = 180° (angles on a straight line) 5x = 100° x = 20° 80° 2x 3x z = 65° (alternate angles) y + 110° = 180° (interior angles)

y = 70° x= y +z = 70° + 65° = 135° x y z 65° 110° 110° n m

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1 Which of the following diagrams shows line PQ parallel to line RS? A C _B _D 2 DIAGRAM 1

In Diagram 1, PQRS is a straight line. Which of the following istrue_?

A x = y _{B x = z} C y = z D x + y + z = 180 3 DIAGRAM 2

In Diagram 2, PRS and QRT are straight lines. Find the value of x.

A 35 B 40 _C ₅₀ Q S U T R 40° 60° x ° P x ° z ° 50° ₁₂₀ ° y ° Q P R S P R Q S 70° 70° P R Q S 70° 70° P R Q S 70° 70° P R Q S 70° 70° 4 DIAGRAM 3

In Diagram 3, CDE, ADF and GDH are straight lines. Find the value of y .

_{A 60} _{C 70}

B 65 D 80

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DIAGRAM 4

In Diagram 4, JK is parallel to MN . Find the value of x.

A 175 C 215

B 185  _{D 225} 6

DIAGRAM 5

In Diagram 5, PRT and QRS are straight lines. Find the value of x.

_{A 75} _{C 95} B 85 D 105 7 100° A B C D 80° 55° 80° 125° J L P S U Q M K T R _ P Q T S x ° R 30° 45° M L 65° 110° K x ° N J G y ° D C F H E B A 110° 50° Answer all questions.

Paper 1

PMR Practice 1

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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? 8

DIAGRAM 7

In Diagram 7, PQRS is a straight line. Find the value of y .

A 50 _{C 60}

B 55 D 70

9

DIAGRAM 8

In Diagram 8, JMN is parallel to KL. Find the value of x.

A 65 C 80

_{B 70} _{D 85}

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DIAGRAM 9

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

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DIAGRAM 10

In Diagram 10, JKL and MNP are straight lines. Which of the angles A_, B_, C _or D_{, is}

equal to x? J x ° K A D C B L M N P  A B C D 30° x ° F E 50° 135° x ° J M L K N y ° P 50° 70° Q R S 12 DIAGRAM 11

In Diagram 11, JLN and KLM are straight lines. Find the value of x.

A 85  _{C 95}

B 90 D 105

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DIAGRAM 12

In Diagram 12, ABC and DEFG are straight lines. Find the value of y .

_{A 50} _{C 60}

B 55 D 70

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DIAGRAM 13

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}

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DIAGRAM 14

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} J K L M 120° ₄₅ ° 15° 30° 20° P Q R S I II III IV V VI 65° 35° 45° 30° 20° 30° P Q R S A D E F G y ° 75° 125° C B J L M N K x ° 35° 60° 2006

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20

DIAGRAM 19

In Diagram 19, JKL is a straight line. Find the value of y .

A 180 C 195

B 190  _{D 205} 21

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?

22

DIAGRAM 21

In Diagram 21, JKL is a straight line. Find the value of x.

A 55 C 75

B 70  _{D 80}

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DIAGRAM 22

In Diagram 22, PQR is a straight line. Find the value of x. A 90 C 130 P x ° Q R 25° 115° 45° 125° x ° J K L P A B D C R K M S x ° Q J L  J K 115° L y ° 16 DIAGRAM 15

In Diagram 15, PQR and SQT are straight lines. Find the value of x.

A 55 C 40

B 45 _{D 35}

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DIAGRAM 16

In Diagram 16, PQR is a straight line. Find the value of y .

A 45 _{C 55}

B 50 D 65

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DIAGRAM 17

In Diagram 17, JKLM is a straight line. Find the value of x.

_{A 10} _{C 20}

B 15 D 26

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DIAGRAM 18

In Diagram 18, PQ and RS are straight lines. Find the value of x + y .

A 90 _{C 180} P y ° Q S R x ° 13 cm 1 2 c m 5 c m J K L M 130° 2x ° 3_x° y ° Q P R 135° 100° 85° 50° P S x ° R T Q 2004

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24

DIAGRAM 23

In Diagram 23, QRS is a straight line and QP = QR. Find the value of x.

A 70  _{C 140}

B 130 D 150

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DIAGRAM 24

In Diagram 24, PQR is a straight line. Find the value of y .

A 35 C 55

_{B 50} _{D 60}

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DIAGRAM 25

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 27

DIAGRAM 26

In Diagram 26, JK, LM and PQ are straight lines. Find the value of x – y .

A 30 C 50 _{B 40} _{D 55} K M y ° x ° Q L J P 70° E F C B q ° p ° s ° r ° A D y ° P 85° 35° Q R P R 110° Q S T x ° 28 DIAGRAM 27

In Diagram 27, JK is parallel to LM . Find the value of x.

A 95  _{C 115}

B 110 D 120

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DIAGRAM 28

In Diagram 28, PQ is parallel to ST . Find the value of x.

_{A 135} _{C 150}

B 145 D 205

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DIAGRAM 29

In Diagram 29, JKL is a straight line. Find the value of x.

A 105 C 120

B 115  _{D 130} 31

DIAGRAM 30

In Diagram 30, JLN and KLM are straight lines. Find the value of x.

A 80  _{C 96} B 84 D 102 M N K J L x ° 56° 40° P M N x ° J K L 235° 75° S T R x ° Q P 80° 125° K J x ° M L 125° 240° 2006

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Answer all questions.

1 In Diagram 1, PQRS is a straight line. Find the value of x. DIAGRAM 1 P Q x ° S R 130°

2 In Diagram 2, PQRS is a straight line. Find the value of x. DIAGRAM 2 x ° P Q R S 70° y °

x + 130° = 180° (angles on a straight line) x = 50°

70° + 70° + y = 180° (sum of angles in∆) y = 40°

x + y = 180° (angles on a straight line) x = 140°

32

DIAGRAM 31

In Diagram 31, BDF is a straight line. Which of the following pairs of lines is parallel? A AB and CD C BC and DE _{B AB and EF} _{D CD and EF} 33

DIAGRAM 32

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} R Q P J K L M 30° 20° 20° 25° 110° B D F E C A 70° 50° 65° 60° 60° 34 DIAGRAM 33

In Diagram 33, JKL is a straight line. Find the value of x + y .

_{A 100} _{C 120}

B 110 D 135

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DIAGRAM 34

In Diagram 34, KL, MN , PQ , RS and TU are straight lines. The value of x + y is the same as A a + d  _{C b + e} B a + f D c + f a b c y f e d x N M Q T R P _S U L K y ° x ° J K L 20° 70° 50° Answer: ______________50° Answer: ______________140°

Paper 2

2005

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4 In Diagram 4, DEF is a straight line. Find the value of x.

DIAGRAM 4

6 In Diagram 6, JKL and LMN are straight lines. Find the value of y .

8 Diagram 8 shows

two parallel lines. Find the value of x. DIAGRAM 8 10 In Diagram 10, JK is parallel to MN . Find the value of x. DIAGRAM 10

65° + 65° + y = 180° (sum of angles in∆) 130° + y = 180°

y = 50°

x = y = 50° (corresponding angles)

D E F 65° x ° y ° 50° + y + 85° = 180° (angles on a straight line) y + 135° = 180° y = 45° 2003

y + 90° + 125° = 360° (angle of one whole turn) y + 215° = 360° y = 145° Answer: ______________225° Answer: ______________50° Answer: ______________315° Answer: ______________45° Answer: ______________30° Answer: ______________230° Answer: ______________145° Answer: ______________65° 3 In Diagram 3, JK is parallel to NM . Find the

value of x.

DIAGRAM 3

5 In Diagram 5,

PQ is parallel to RS. Find the value of x + y .

DIAGRAM 5

7 In Diagram 7,

DEF and EGH are straight lines. Find the value of x.

9 In Diagram 9,

JK is parallel to LM . Find the value of y .

DIAGRAM 9

45° + x + y = 360° (angle of one whole turn) x + y = 360° – 45° = 315° M L x ° K J _N 60° 105° y ° L M N J K 50° 85° D E F G H 50° 70° x ° DIAGRAM 7 x + 90° + 60° = 180° (interior angles) x + 150° = 180° x = 30° DIAGRAM 6

60° + 75° + x = 360° (angle of one whole turn)

x + 135° = 360° x = 225°

x + 60° + 70° = 360° (angle of one whole turn) x + 130° = 360°

x = 230°

60° + x + 235° = 360° (angle of one whole turn) x + 295° = 360° x = 65° 235° 120° N M J K L x ° y ° J K L 125° M x ° 120° 110° x ° Q P R y ° S 135°

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12 In Diagram 12, JK is parallel to MN . Find the value of x.

DIAGRAM 12

14 In Diagram 14, AB, CD and EF are straight lines. Find the value of x.

DIAGRAM 14 E F B 2_x° + 30° 70° A D C 4x + 15° = 50° + 65° 4x = 115° – 15° 4x = 100° x = 25° M N L 4_x° + 15° 50° 115° K J

11 In Diagram 11, PQ is parallel to ST . Find the value of x.

DIAGRAM 11

13 In Diagram 13,

RSTU is a straight line. Find the value of x and of y.

DIAGRAM 13 DIAGRAM 15 T U 65° y ° + 15° x ° Q S P R 2x = 4x – 60° (alternate angles) 2x = 60° x = 30°

y = 2x (vertically opposite angles) = 2 ×30° = 60° P S Q R x ° T 105° 120° 60° + x + 75° = 180° (sum of angles in ∆) x + 135° = 180° x = 45° x = 65° (alternate angles)

x + y + 15° = 180° (angles on a straight line) 65° + y + 15° = 180°

y = 100°

15 In Diagram 15, PQ, RS and TU are straight lines. Find the value of y .

Answer: _____________________45° Answer: ______________25°

Answer: _____________________x = 30°, y = 60° Answer: ______________40°

Answer: _____________________100°

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.

DIAGRAM 16 T S U x ° 125° y ° R P Q

y + 55° + 55° = 180° (sum of angles in ∆) y = 70°

x + y = 180° (interior angles) x = 110° Answer: ______________110° 70° + 2x + 30° = 180° (angles on a straight line) 2x = 180° – 100° x = 40° R y ° S 2x ° 4_x°– 60° T U