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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