Basic properties of circles - (10.4)

Understand the tests for concyclic points and cyclic quadrilaterals

• In 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.
• In 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)

• if a pair of opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic
opp angles supp
• In 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.
• In 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. \angle s supp.)
• if the exterior angle of a quadrilateral equals its interior opposite angle, then the quadrilateral is cyclic
(ext. $$\angle\, =$$ int. opp. $$\angle s$$)
• In 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.
$$\begin{array}{1}\angle ADE + \angle DAE + \angle AED = 180^\circ \\\angle ADE + 41^\circ + 62^\circ = 180^\circ \\\angle ADE = 77^\circ \end{array}$$ ( sum of )
$$\therefore \angle ADE = \angle BCE = 77^\circ$$
$$\therefore$$ BCED is a cyclic quadrilateral. (ext.   int. opp. )