In the figure below, if A+B+C+D+E+F+G=x degrees, then what is x?
A B C D E F G


Answer:

540

Step by Step Explanation:
  1. Let us divide the shape into triangles with respect to a center O.
    A B C D E F G O
  2. 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.
  3. Now, A+B+C+D+E+F+G=x
    Sum of the angles of the 7 triangles Angle formed around O=x
    7×1802×360=x
    x=540
  4. Hence, the value of x is 540.
  5. 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)]=1260A+B+C+D+E+F+G+[2AOG+2GOF+2FOE+2EOD+2DOC+2COB+2BOA]=1260A+B+C+D+E+F+G+2[AOG+GOF+FOE+EOD+DOC+COB+BOA]=1260A+B+C+D+E+F+G+(2×360)=1260A+B+C+D+E+F+G=1260720A+B+C+D+E+F+G=540

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