Question

Find <ABC. Please help.​

Answer 1

Answer:

$\angle\ ABC=70$

Step-by-step explanation:

$We\ are\ given\ that,\\AB\ and\ AC\ are\ two\ equal\ chords\ on\ the\ circle\ ABC.\\ Let\ the\ center\ of\ the\ circle\ ABC\ be\ O.\\ Angle\ subtended\ at\ the\ Center\ of\ the\ circle\ by\ Chord\ AB\ is\ 110.\\Hence,\\Lets\ connect\ AC\ to\ form\ a\ triangle- \triangle ABC.\\Also,\\O\ forms\ the\ Centroid\ of\ \triangle ABC.$

$By\ joining\ sides\ AO\ and\ OB,\ we\ obtain\ \triangle ABO.\\Hence,\\\angle BOA=110\ \\Now,\\Considering\ point\ C\ on\ Circle\ ABC,\ lets\ consider\ \angle BCA.$

$We\ know\ that,\\'Angle\ subtended\ by\ a\ chord\ at\ the\ center\ is\ double\ the\ angle\\ subtended\ by\ the\ same\ chord\ at\ the\ respective\ arc\ of\ the\ circle'.\\Here,\\As\ BOA\ is\ an\ angle\ subtended\ by\ the\ chord\ AB\ at\ the\ center\ of\ circle\ ABC\\ while,\ BCA\ is\ an\ angle\ subtended\ by\ the\ same\ chord\ AB\ at\ the\ major\ arc\ of\\ the\ Circle\ ABC.$

$Here,\\\angle BOA=2 \angle BCA\\Hence,\\Substituting\ \angle BOA=110,\\110=2*\angle BCA\\Hence,\\\angle BCA=\frac{110}{2}=55$

$Now,\\We\ also\ know\ that,\\'Base\ angles\ opposite\ to\ equal\ sides\ are\ equal\ too'.\\Here,\\In\ \triangle ABC,\\As\ BA=BC,\\\angle BAC= \angle BCA\\\\\therefore \angle BAC= \angle BCA=55$

$The\ Angle\ Sum\ Property\ of\ a\ Triangle\ States\ that,\\'The\ Sum\ of\ all\ Interior\ Angles\ of\ a\ Triangle\ is\ 180.'\\Hence,\\As\ Polygon\ ABC\ is\ a\ triangle,\\\angle BCA +\angle BAC +\angle ABC=180\\Substituting\ \angle BCA=\angle BAC=55,\\55+55+ \angle ABC=180\\Hence,\\110+ \angle ABC=180\\Or,\\\angle ABC=180-110=70$