CENTRAL ANGLES AND ARC MEASURES

Example 1 : 

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

(i) m∠arc BC

(ii) m∠arc ABC

Solution :

(i) m∠arc BC :

AB is the diameter of the above circle. 

m∠arc AB = 180°

m∠arc BC + m∠arc CA = 180°

m∠arc BC + 123° = 180°

m∠arc BC = 57°

(ii) m∠arc ABC :

m∠arc ABC = m∠arc AB + m∠arc BC

= 180° + 57°

= 237°

Example 2 :

From the diagram shown above, find the following measures. 

(i) m∠arc CD

(ii) m∠AOC

(iii) m∠arc BD

(iv) m∠arc ABC

(v) m∠arc CBD

Solution :

(i) m∠arc CD :

m∠AOB and m∠COD are vertical angles. 

m∠COD = m∠AOB

m∠arc CD = m∠arc AB

m∠arc CD = 55°

(ii) m∠AOC :

BC is the diameter of the above circle. 

m∠arc BAC = 180°

m∠arc BA + m∠arc AC = 180°.

55° + m∠arc AC = 180°.

m∠arc AC = 125°.

m∠AOC = 125°.

(iii) m∠arc BD : 

m∠BOD and m∠AOC are vertical angles. 

m∠BOD = m∠AOC

m∠BOD = 125°

m∠arc BD = 125°

(iv) m∠arc ABC : 

m∠arc ABC = m∠arc ABD + m∠arc DC

= 180° + 55°

= 235°

(v) m∠arc CBD : 

m∠arc CBD = m∠arc CAB + m∠arc BD

= 180° + 125°

= 305°

Example 3 :

Find the value of x in the diagram shown below. 

From the diagram shown above, find the  m∠arc QTR.

Solution :

Find m∠arc QP :

PS is the diameter of the above circle.

m∠arc PTS = 180°

m∠arc PT + m∠arc TS = 180°

135° + m∠arc TS = 180°

m∠arc TS = 45°

Find m∠arc QTR :

m∠QTR = m∠arc QT + m∠arc TS + m∠arc SR

= 180° + 45° + 81°

= 306°

Example 4 :

From the diagram shown above, find the following measures. 

m∠BOD, m∠BOE and m∠BOC

Solution :

Find m∠BOD :

In the circle above,

m∠arc AB + m∠arc BCD + m∠arc DE + m∠arc EA = 360°

60° + m∠arc BCD + 86° + 154° = 360°

m∠arc BCD + 300° = 360°

m∠arc BCD = 60°

m∠BOD = 60°

Find m∠BOE :

m∠BOE = m∠arc BCD + m∠arc DE

= 60° + 86°

= 146°

Find m∠BOC :

In the above diagram, m∠BOC = m∠COD.

m∠BOC + m∠COD = m∠BOD

m∠BOC + m∠BOC = m∠BOD

2m∠BOC = 60°

m∠BOC = 30°

Example 5 :

From the diagram shown above, find the following measures. 

m∠KOL and m∠arc MNK

Solution :

In the diagram above, m∠JON and ∠KOM are vertical angles.

m∠KOM = m∠KOM

m∠KOM = 126°

m∠KOL + m∠LOM = 126°

In the above diagram, m∠KOL = m∠LOM.

m∠KOL + m∠KOL = 126°

2m∠KOL = 126°

m∠KOL = 63°

Find m∠arc MNK :

m∠arc MNK = 360° - m∠arc KLM

m∠arc MNK = 360° - m∠KOM

m∠arc MNK = 360° - 126°

m∠arc MNK = 234°

Example 6 :

Find the measure of each arc.

Find the arc measures of following :

a) JL

b) KM

c) JLM

d) JM

Solution :

a)

m∠arc JL = m∠arc JK +  m∠arc KL

= 53 + 79

m∠arc JL = 132

b)  

m∠KM =  m∠arc KL +  m∠arc LM

= 79 + 68

m∠KM = 147

c)  

m∠JLM =  m∠arc JK + m∠arc KL +  m∠arc LM

= 53 + 79 + 68

= 200

d)

m∠JM = 360 - m∠arc JLM

= 360 - 200

= 160

Example 7 :

In the circle K is the center. Find the measure of arc DE and arc BA

Solution :

m∠arc BC = 30

m∠arc BA = ?

m∠arc AE = 55

m∠arc BC + m∠arc BA + m∠arc AE = 180

30 + m∠arc BA + 55 = 180

m∠arc BA = 180 - 85

m∠arc BA = 95

m∠arc DE + m∠arc AE = 180

m∠arc DE + 55 = 180

m∠arc DE = 180 - 55

m∠arc DE = 125

Example 8 :

A recent survey asked teenagers whether they would rather meet a famous musician, athlete, actor, inventor, or other person. The circle graph shows the results. Find the indicated arc measures.

a)  m∠arc AC

b)  m∠arc ACD

c)  m∠arc ADC

d)  m∠arc EBD

Solution :

a)  m∠arc AC = m∠arc CB +  m∠arc BA

= 108 + 29

= 137

b)  m∠arc ACD = m∠arc AC + m∠arc DC

= 137 + 83

= 220

c)  m∠arc ADC = m∠arc AE + m∠arc ED + m∠arc DC

= 79 + 61 + 83

= 223

d)  m∠arc EBD = 360 - m∠arc DE

= 360 - 61

= 299

Example 9 :

Find the value of x. Then find the measure of arc AB

Solution :

m∠arc AB + m∠arc BC + m∠arc CD + m∠arc DA = 360

5x + 40 + 70 + 75 + 2x = 360

185 + 7x = 360

7x = 360 - 185

7x = 175

x = 25

m∠arc AB = 5x

= 5(25) 

= 125

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