ANGLES AND PARALLEL LINES

1. In the diagram, QTR is a straight line and PQ is parallel to SR.

Given that TR = SR , .

calculate

(a) the value of x,

(b) the value of y,

(c) the value of z.

Answer: (a) x = ___________________0 [1]

(b) y = ___________________0 [1]

(c) z = ___________________0 [2]

z0

400

S

x0y0

350

250

RT

P

Q

2. In the rhombus ABCD, DB cuts AC at X and ∠DAC=50 ° . The point P on AD is such that PX = AX. The line PX produced meets BC at Q. Calculate

(a) ∠ AXP,

(b) ∠BQP,

(c) ∠ ADC .

Answer (a )∠ AXP=…………………………… o [1]

(b)∠BQP= .…………………………… o [2]

Q

C

XP

D

BA

50o

E F G

H J

K 81

61

(c)∠ ADC=…………… ……… ………o[2]

2. Refer to the triangle shown below. Form an equation in x and solve for x.

Answer x =……………………… o [2]

3. In the diagram below, JFE is an isosceles triangle and GFJH is a rhombus. Given that

FHJ=61∘, JKF = 81˚, EF = FJ and both EFG and EKJ are straight lines,

find (a) FGH, (b) KJH, (c) EFK.

x + 32°

3x + 16°

Answer (a)……………………………… [1]

(b)……………………………… [1]

(c)………………………………. [1]

4. In the figure below AB // CD and EFG and HFI are straight lines.

Find the values of p, q, r and s.

Answer: p = o [1][1]

q = o

r = o [1]

s = o [1]

5. In the figure, given that lines PQ and AC are parallel and lines AQ and BY are parallel. If angle PQA = 55o and angle QXB = 98o, find the unknown angles x, y and z.

Answer: x = _____________ [1]

y = _____________ [1]

z = _____________ [1]

6. In the diagram below, the straight line ABC is parallel to WX, and BY is parallel to CX. ABY = 57°, XWY = 132° and BXC = 63°. Calculate(a) BCX,(b) BXW.(c) BYW. 132o

63o

W X

A B C

Y

57o

Answer: (a) ________________ [1]

(b) ________________ [1]

(c) ________________ [2]

7. In the diagram, ABDE is a rhombus.  C is a point on DB produced such that AB = AC and.

Find

(a) ,

(b) .

 

 

Answer (a) [2]

(b) [2]

8. In the diagram, BCD is a straight line and DE is parallel to BA.  It is given

that BC = CA, .

Calculate(a) the value of x, [2]

(b) the value of y. [3]

25

b

a305°

A

BC

D

E

F

9. In the given figure, BXC is a straight line, AX = XC, ∠AXC = 126° and ∠ABC = 58°.

Calculate the value of

(a) ∠CAX,

(b) ∠BAX,

State the reasons clearly.

Answer (a) ……………………………….. [2]

(b) ……………………………….. [2]

10. Stating all reasons clearly, find angles ∠a and ∠b.

A

XB58 126

C

Answer ∠a ……………………..……………...[2]

∠b ………………….….......................[3]

11. In the figure, ABCD is a parallelogram and ADE is an isosceles triangle.

Calculate

(a) ∠ ABC ,

(b) ∠DAE ,

(c) ∠ AEF .

Answer (a) ∠ ABC = o [1]

(b) ∠DAE = o [2]

(c) ∠ AEF = o [1]

12. In the diagram, PQ is parallel to ST. Reflex ∠PQR=230 ° and ∠RST=15° .Calculate the value of z. [3]

C D E F

56o

15

P

T

Q

S

230R

z

13. In the diagram, PS is parallel to QR. SPU = 20, PSU = 65 and PQR = 130. Calculate(a) QPU(b) SRT(c) PUR

R

TU

S

QP

65o

20o 130o

Answer: (a) ___________________ [2]

(b) __________________ [1]

(c) __________________ [1]

14.

In the diagram above, ABC is a triangle in which AC = BC. The point D is on

AC produced and DE is parallel to CB. Given that ∠BCD=108° , calculate x, y

and z, stating the reasons for your working.

A

B

C D

E

F

yo xo

zo

108o

Answer: x = __________o [1]

y = __________o [1]

z = __________o [1]

15. In the diagram, the lines AD and BC meet at X. AB is parallel to CD and BY is parallel to AD.

Given ∠XBY =67 ° and ∠XDC=55° ,

calculate

(a) ∠ AXB ,

(b) ∠ ABX .

oAnswer (a) ∠ AXB = …………..….…. [1] o

(b) ∠ ABX = …………..….…. [2]

16. In the diagram, ABC is a triangle in which AC = BC. The point D is on AC produced and DE is parallel to CB. Given that ∠BCD = 60°, calculate the value of(a) x,(b) y,(c) z.

A

B

C

D

X

Y

67

55

Ans: (a) _________________ [1]

Ans: (b) __________________ [1]

Ans: (c) __________________ [1]

17. In the diagram, ABCD is a parallelogram. Given that ∠ ADC=58 ° , ∠CBD=37 ° and

∠BMC=60 ° . Stating your reasons clearly, calculate

(a) ∠ ABC ,

(b) ∠CAB .

Answer: (a) ________________° [1]

(b) ________________° [2]

18. Find the value of x and of y in the diagram below.

M

C

B

58°

37°

60°

D

A

xo

44o

yo

Ans:(a) x =_________; y =_________ [4]

19. In the diagram, ABC is parallel to GFE, AF is parallel to DE and BF is parallel to CD.

GFA = 55o and CDE = 130o.

Calculate (state all geometrical reasons)

(a) DEF,

(b) FBC.

Answer (a) ________________[1]

(b) ________________[3]

20. In the following diagram, find the values of a and b.

G

55o

130o

E

D

CB

F

A

37o

b a

42o

Answer: a = ________________ [1]

b = ________________ [1]21. In the diagram below, the lines AB and CDEF are parallel.

By stating your reasons clearly, find the value of

(a) x,

(b) y,

(c) z,

(d) w.

w

3z

147°°

x y

72°

F

BA

Ans: (a) x = _________________________ [1]

(b) y = _________________________ [1]

(c) z = _________________________ [1]

(d) w = _________________________ [1]

22. Find the values of x and y. Show your workings and statements clearly. A

B

Answer x = ° [2]

y = ° [1]

23. In the diagram below, BEF is an isosceles triangle with BE = BF and BFE 70. Given also that AB is parallel to DEF, DC is parallel to EB and AC is parallel to BF, calculate

(a) ∠BAC , [2]

(b) ∠CDE , [2]

(c) reflex∠ ACD . [3]

C

C D