Answer:
[tex]x=36[/tex]
Step-by-step explanation:
[tex]We\ are\ given:\\ABCDE\ is\ a\ regular\ pentagon.\\Hence,\\The\ word\ regular\ simply\ means\ that\ all\ angles\ and\ sides\ of\ the\ polygon\\ are\ equal.\\Hence,\\AB=BC=CD=DE=EA\ and\ \angle ABC= \angle BCD\ = \angle CDE\ = \angle DEA= \angle EAB.\\Hence,\\We\ know\ that,\\The\ Interior\ Angle\ Sum\ Property\ Of\ A\ Polygon\ with\ n\ sides,\\Interior\ Angle\ Sum\ Property=180(n-2)\\[/tex]
[tex]As\ we\ know\ that,\\The\ pentagon\ has\ 5\ sides,\\The\ Interior\ Angle\ Sum\ Property\ Of\ A\ Pentagon=180(5-2)=180*3=540.\\Now,\\\angle ABC+ \angle BCD\ + \angle CDE\ + \angle DEA+ \angle EAB=540\\As\ we\ already\ know\ that,\\\angle ABC= \angle BCD\ = \angle CDE\ = \angle DEA= \angle EAB.\\Hence,\\5\ \angle BCD=540\\\angle BCD=\frac{540}{5} =108\\[/tex]
[tex]Hence,\\Now\ lets\ consider\ the\ \triangle BCD.\\Hence,\\We\ know\ that\ as\ ABCDE\ is\ a\ regular\ pentagon,\\BC=CD.\\We\ know\ that,\\"The\ base\ angles\ opposite\ to\ equal\ sides\ are\ equal."\\As\ BC=CD\ in\ \triangle BCD,\\\angle DBC= \angle BDC.\\Hence,\\\angle DBC= \angle BDC=x\\[/tex]
[tex]Now,\\ The\ angle\ sum\ property\ of\ triangle\ tells\ us\ that:\\"The\ sum\ of\ all\ interior\ angles\ in\ a\ triangle\ is\ 180."\\Hence\ in\ \triangle BCD,\\\angle BCD+ \angle BDC + \angle DBC=180\\Hence,\\x+x+108=180\\2x+108=180\\2x=180-108\\2x=72\\x=\frac{72}{2}=36[/tex]
