# CENTRAL ANGLES

## Central angles

Central angles :

The central angle of a circle is 360 degree.Here we are going to see some example problems.

Example 1 :

What is  ∠PSQ?

Solution :

Since the central-angle of a circle is 360 degree,

∠PSQ + ∠QSR + ∠RSP  =  360°

∠PSQ + 80° + 140°  =  360°

∠PSQ + 120°  =  360°

∠PSQ   =  360° -  120°

∠PSQ   =  240°

Hence the required angle is 240°.

Example 2 :

What is  ∠UVT?

Solution :

Since the central angle of a circle is 360 degree,

∠SVU + ∠UVT + ∠TVS  =  360°

140° + ∠UVT + 130°  =  360°

270° + ∠UVT  =  360°

∠UVT  =  360° - 270°

∠UVT  =  90°

Hence the required angle is 90°.

Example 3 :

What is  ∠UWT?

Solution :

Since the central-angle of a circle is 360 degree,

∠VWU + ∠UWT + ∠TWV  =  360°

130° + ∠UWT + 80°  =  360°

210° + ∠UWT   =  360°

∠UWT  =  360° - 210°

∠UWT  =  150°

Hence the required angle is 150°.

Example 4 :

What is  ∠AOE and ∠EOD ?

Solution :

Since AD is a straight line, the measure of angle AOD is 180 degree

∠AOE + EOD  =  180°

x + 138° + x + 48°  =  180°

2x + 186°  =  180°

By subtracting 186° on both sides, we get

2x + 186° - 186°  =  180° - 186°

2x  =  180° - 186°

2x  =  -6

Divide by 2 on both sides, we get

x  =  -3

∠AOE  =  -3 + 138  =  135°

∠EOD  =  -3 + 48  =  45°

Hence the required angles are 135° and 45°.

Example 5 :

What is  ∠FDG ?

Solution :

∠FDE + ∠EDG + ∠FDG  =  360°

70 + 140 ∠FDG  =  360°

210 ∠FDG  =  360°

Subtract 210 on both sides, we get

210 ∠FDG - 210  =  360° - 210

∠FDG  =  150°

Example 6 :

What is  ∠SRU ?

Solution :

∠SRU + ∠URT + ∠TRS  =  360°

∠SRU + 130 + 80  =  36

∠SRU + 210  =  36

Subtract 210° on both sides

∠SRU + 210° - 210°  =  360° - 210

∠SRU  =  15

Example 7 :

What is  ∠SPQ ?

Solution :

∠SPR + ∠RPQ + ∠QPS  =  360°

130 + 110 + ∠QPS  =  360°

240 +  ∠QPS  =  360°

Subtract 240 on both sides

240°  + ∠QPS - 240°  =  360° - 240°

∠QPS  =  120°

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