In the figure below, if ∠A+∠B+∠C+∠D+∠E+∠F+∠G=x degrees, then what is x?
Answer:
540
- Let us divide the shape into triangles with respect to a center O.
- We observe, from the diagram, that seven triangles have formed e.g.OAC,OBD, etc. We also observe that the angles around O have been added twice.
- Now, ∠A+∠B+∠C+∠D+∠E+∠F+∠G=x
⟹ Sum of the angles of the 7 triangles − Angle formed around O=x
⟹7×180−2×360=x
⟹x=540 - Hence, the value of x is 540.
- Illustration of step 3:
According to the angle sum property of a triangle, the sum of angles of a triangle is 180∘.
For 7 triangles AOF,GOE,FOD,EOC,DOB,COA,BOG,
(∠FAO+∠OFA+∠AOF)+(∠GOE+∠OEG+∠EGO)+(∠FOD+∠OFD+∠ODF)+(∠EOC+∠OEC+∠OCE)+(∠DOB+∠ODB+∠OBD)+(∠COA+∠OCA+∠OAC)+(∠BOG+∠OGB+∠OBG)=7×180
Now,
[(∠CAO+∠OAF)+(∠DBO+∠OBG)+(∠ECO+∠OCA)+(∠FDO+∠ODB)+(∠GEO+∠OEC)+(∠AFO+∠OFD)+(∠EGO+∠OGB)]+[(∠AOG+∠GOF)+(∠GOF+∠FOE)+(∠FOE+∠EOD)+(∠EOD+∠DOC)+(∠DOC+∠COB)+(∠COB+∠BOA)+(∠BOA+∠AOG)]=1260∘⟹∠A+∠B+∠C+∠D+∠E+∠F+∠G+[2∠AOG+2∠GOF+2∠FOE+2∠EOD+2∠DOC+2∠COB+2∠BOA]=1260∘⟹∠A+∠B+∠C+∠D+∠E+∠F+∠G+2[∠AOG+∠GOF+∠FOE+∠EOD+∠DOC+∠COB+∠BOA]=1260∘⟹∠A+∠B+∠C+∠D+∠E+∠F+∠G+(2×360∘)=1260∘⟹∠A+∠B+∠C+∠D+∠E+∠F+∠G=1260∘−720∘⟹∠A+∠B+∠C+∠D+∠E+∠F+∠G=540∘