Basic properties of circles  (10.4)
Understand the tests for concyclic points and cyclic quadrilaterals

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Theoryif A and D are two points on the same side of the line BC and ∠BAC = ∠BDC , then A , B , C and D are concyclic
converse of s in the same segment) 
ExamplesIn the figure, AEC and BED are straight lines. It is given that \(\angle BDC = 30^\circ \),
\(\angle ACB = 50^\circ \) and \(\angle ABC = 100^\circ \). Prove that A, B, C and D are concyclic. 
SolutionsIn ABC,
\(\begin{array}{1}\angle BAC + \angle ABC + \angle ACB = 180^\circ \\\angle BAC + 100^\circ + 50^\circ = 180^\circ \\\angle BAC = 30^\circ \end{array}\) ( sum of )
\(\therefore \angle BAC = \angle BDC = 30^\circ \)
\(\therefore\) A, B, C and D are concyclic. (converse of s in the same segment)

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Theoryif a pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic
opp angles supp 
ExamplesIn the figure, AEC and BED are straight lines. It is given that \(\angle BAD = 82^\circ \),
\(\angle CBD = 32^\circ \) and \(\angle BDC = 50^\circ \). Prove that A, B, C and D are concyclic. 
SolutionsIn BCD,
\(\begin{array}{1}\angle BCD + \angle CBD + \angle BDC = 180^\circ \\\angle BCD + 32^\circ + 50^\circ = 180^\circ \\\angle BCD = 98^\circ \end{array}\) ( sum of )
\(\begin{array}{1}\angle BAD + \angle BCD = 82^\circ + 98^\circ \\ = 180^\circ \end{array}\)
\(\therefore\) A, B, C and D are concyclic. (opp. s supp.)

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Theoryif the exterior angle of a quadrilateral equals its interior opposite angle, then the quadrilateral is cyclic
(ext. int. opp. ) 
ExamplesIn the figure, D and E are points on AB and AC respectively. \(\angle DAE = 41^\circ \),
\(\angle AED = 62^\circ \) and \(\angle ACB = 77^\circ \). Prove that BCED is a cyclic quadrilateral. 
SolutionsIn ADE,
\(\begin{array}{1}\angle ADE + \angle DAE + \angle AED = 180^\circ \\\angle ADE + 41^\circ + 62^\circ = 180^\circ \\\angle ADE = 77^\circ \end{array}\) ( sum of )
\(\therefore \angle ADE = \angle BCE = 77^\circ \)
\(\therefore\) BCED is a cyclic quadrilateral. (ext. int. opp. )