# CENTRAL ANGLES AND ARC MEASURES WORKSHEET

Problem 1 :

From the diagram shown above, find the following arc measures.

(i)  m∠arc RS

(ii)  m∠arc SRT

Problem 2 :

From the diagram shown above, find the following measures.

(i)  m∠arc ML

(ii)  m∠JNM

(iii)  m∠arc KL

(iv)  m∠arc JKM

(v)  m∠arc MKL

Problem 3 :

From the diagram shown above, find the following measures.

(i)  m∠UXV

(ii)  m∠arc ST

(iii)  m∠arc WV

(iv)  m∠arc TW

(v)  m∠arc TVW

Problem 4 :

From the diagram shown above, find the following measures.

(i)  m∠arc CD

(ii)  m∠arc FD

(iii)  m∠arc DCF

(iv)  m∠arc GDF

Problem 5 :

Find the value of x in the diagram shown below.

Problem 6 :

Find the value of x in the diagram shown below.

Problem 7 :

Find the value of x in the diagram shown below.

Problem 8 :

Find m∠arc EF in the diagram shown below.

Problem 9 :

From the diagram shown above, find the following arc measures.

(i)  m∠arc VW

(ii)  m∠arc UXV

Problem 10 :

In the diagram shown above,

m∠arc VQ  =  (y + 7)°

m∠arc QR  =  (x + 11)°

m∠arc RS  =  (3y)°

m∠arc ST  =  65°

Find the values of x and y.

(i) m∠arc RS :

RT is the diameter of the above circle.

m∠arc RT = 180°

m∠arc RS + m∠arc ST = 180°

m∠arc RS + 122° = 180°

m∠arc RS = 58°

(ii) m∠arc SRT :

m∠arc SRT = m∠arc SR + m∠arc RT

= 58° + 180°

= 238°

(i) m∠arc ML :

m∠MNL and m∠JNK are vertically opposite angles.

m∠MNL = m∠JNK

m∠arc ML = m∠arc JK

= 21°

(ii) m∠JNM :

MK is the diameter of the above circle.

m∠MK = 180°

m∠JNM + m∠JNK = 180°

m∠JNM + 21° = 180°

= 159°

(iii) m∠arc KL :

m∠KNL and m∠JNM are vertically opposite angles.

m∠KNL = m∠JNM

m∠KNL = 159°

m∠arc KL = 159°

(iv) m∠arc JKM :

m∠arc JKM = m∠arc JK + m∠arc KM

m∠arc JKM = 21° + 180°

= 201°

(v) m∠arc MKL :

m∠arc MKL = m∠arc MK + m∠arc KL

= 180° + 159°

= 339°

(i) m∠UXV :

m∠UXV and m∠SXW are vertically opposite angles.

m∠UXV = m∠SXW

= 72°

(ii) m∠arc ST :

WU is the diameter of the above circle.

m∠arc WU = 180°

m∠arc WS + m∠arc ST + m∠arc TU = 180°

72° + m∠arc ST + 87° = 180°

m∠arc ST + 159° = 180°

m∠arc ST = 21°

(iii) m∠arc WV :

SV is the diameter of the above circle.

m∠arc SV = 180°

m∠arc SW + m∠arc WV = 180°

72° + m∠arc WV = 180°

m∠arc WV = 108°

(iv) m∠arc TW :

m∠arc TW = m∠arc TS + m∠arc SW

m∠arc TW = 21° + 72°

m∠arc TW = 93°

(v) m∠arc TVW :

m∠arc TW + m∠arc TVW = 360°

93° m∠arc TVW = 360°

m∠arc TVW = 267°

(i) m∠arc CD :

In the diagram shown above, m∠CHD is a right angle.

m∠arc CD = 90°

(ii) m∠arc FD :

m∠arc FD = m∠arc FE + m∠arc ED

m∠arc FD = 34° + 90°

m∠arc FD = 124°

(iii) m∠arc DCF :

m∠arc DCF + m∠arc FD = 360°

m∠arc DCF + 124° = 360°

m∠arc DCF = 236°

(iv) m∠arc GDF :

m∠arc GDF = m∠arc GD + m∠arc DF

= 180° + 124°

= 304°

RT is the diameter of the above circle.

m∠arc RT = 180°

m∠arc RS + m∠arc ST = 180°

(15x - 26)° + 26° = 180°

15x - 26 + 26 = 180

15x = 180

x = 12

GE is the diameter of the above circle.

m∠arc GE = 180°

m∠arc GF + m∠arc FE = 180°

(12x - 25)° + (x - 3)° = 180°

12x - 25 + x - 3 = 180

13x - 28 = 180

13x = 208

x = 16

m∠SWT and m∠RWV are vertically opposite angles.

m∠SWT = m∠RWV

m∠arc ST = m∠arc RV

= 45°

RT is the diameter of the above circle.

m∠arc RT = 180°

m∠arc RS + m∠arc ST = 180°

(6x + 33)° + 45° = 180°

6x + 33 + 45 = 180

6x + 78 = 180

6x = 102

x = 17

m∠DHE and m∠GHF are vertically opposite angles.

m∠DHE = m∠GHF

m∠arc DE = m∠arc GF

(15x - 7)° = (13x + 3)°

15x - 7 = 13x + 3

2x = 10

x = 5

Finding m∠arc DE :

m∠arc DE = (15x - 7)°

= [15(5) - 7]°

(75 - 7)°

= 68°

DF is the diameter of the above circle.

m∠arc DF = 180°

m∠arc DE + m∠arc EF = 180°

68° + m∠arc EF = 180°

m∠arc EF = 112°

(i) m∠arc VW :

m∠UZV and m∠YZX are vertically opposite angles.

m∠UZV = m∠YZX

m∠arc UV = m∠arc YX

= 26°

In the diagram shown above, m∠UZW is a right angle.

m∠arc UW = 90°

m∠arc UV + m∠arc VW = 90°

26° + m∠arc VW = 90°

m∠arc VW = 64°

(ii) m∠arc UXV :

m∠arc UXV = m∠arc UX + m∠arc XW + m∠arc WV

m∠arc UXV = 180° + 90° + 64°

m∠arc UXV = 334°

Given :

m∠arc VQ = (y + 7)°

m∠arc QR = (x + 11)°

m∠arc RS = (3y)°

m∠arc ST = 65°

In the diagram shown above, m∠QWS is a right angle.

m∠arc QRS = 90°

m∠arc QR + m∠arc RS = 90°

(x + 11)° + (3y)° = 90°

x + 11 + 3y = 90

x + 3y = 79 ----(1)

And also,

m∠arc VQ + m∠arc ST = 90°

(y + 7)° + 65° = 90°

y + 7 + 65 = 90

y + 72 = 90

y = 18

Substitute 18 for y in (1).

(1)----> x + 3(18) = 79

x + 54 = 79

x = 25

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