# Line segmants ab and cd bisect each other at o. Join ac and ab. Prove that aoc is equal to bod

$By\;linear\;pair\\*\angle AOC+\angle AOD=180^{\circ}\;\;\;\;\;\;\;\;...eq(1)\\* \angle AOD+\angle BOD=180^{\circ}\;\;\;\;\;\;...eq(2)\\*from\;eq(1)\;and \;eq(2)\\*\angle AOC+\angle AOD=\angle AOD+\angle BOD\\*\Rightarrow \angle AOC=\angle BOD\\* Similarly\;(\angle AOD=\angle COB)\\* \therefore \angle AOC=\angle BOD\\* Line\;segments\;AB\;and\;CD\; bisect\;each\;other\;at\;O.\;So,\;O\;is\;midpoint\;of\;CD\;and\;AB\\* \Rightarrow AO=OB\;and\;CO=OD\\*Now,\;In\;\triangle AOC\;and\; \triangle BOD\\*\Rightarrow OA=OB\\*\Rightarrow OC=OD\\*\Rightarrow \angle AOC=\angle BOD\;\;\;\;\;\;(vertically\;opposite\;angle)\\*So,\;by\; SAS\;property\;of\;congruency\;triangles\;are\;congruent\\* \therefore \triangle AOC\cong \triangle BOD$

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