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  1. Chapter 10 Class 9 Circles
  2. Serial order wise
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Transcript

Theorem 10.12 If the sum of a pair of opposite angles of a quadrilateral is 180 , the quadrilateral is cyclic. Given : ABCD is quadrilateral such that BAC + BDC = 180 Prove : ABCD is a cyclic quadrilateral Proof : Since A, B, C are non collinear One circle passes through three collinear points Let us draw a circle C1 with centre at O Let us assume D does not lie on C1 Now, ABCD is cyclic quadrilateral BAC + BD C = 180 But given BAC + BDC = 180 Thus, BD C = BDC Now, In BDD BD C = BDD + DBD BD C = BDC + DBD BDC = BDC + DBD BDC BDC = DBD DBD = 0 D and D Coincides Our assumption was wrong Point D lies on circle C1 A, B, C, D are concyclic. Hence proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.