Question

In figure, rays OA, OB, OC, OD and OE have the common end point 0. Show that ∠AOB +∠BOC +∠COD +∠DOE +∠EOA = 360°.

Answer 1

Given: Rays OA, OB, OC, OD and OE have the common endpoint O. 

Draw an opposite ray OX to ray OA, which make a straight line AX.

From figure: 

∠AOB and ∠BOX are linear pair angles, therefore, 

∠AOB +∠BOX = 180° 

Or, ∠AOB + ∠BOC + ∠COX = 180° —–—–(1)

Also, 

∠AOE and ∠EOX are linear pair angles, therefore, 

∠AOE + ∠EOX = 180° 

Or, ∠AOE + ∠DOE + ∠DOX = 180° —–(2) 

By adding equations, (1) and (2), we get; 

∠AOB + ∠BOC + ∠COF + ∠AOE + ∠DOF + ∠DOE = 180° + 180° 

∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360° 

Hence Proved.