Basic properties of circles  (10.3)
Understand the properties of a cyclic quadrilateral

Theory
the opposite angles of a cyclic quadrilateral are supplementary

ExamplesIn the figure, A, B, C, D and E are points on the circumference. AD intersects
BE at F. If \(\angle BFD = 95^\circ \) and \(\angle ADE = 30^\circ \), find x. 
SolutionsIn∆DEF,
\(\begin{array}{c}\angle DEF + 30^\circ = 95^\circ \\\angle DEF = 65^\circ \end{array}\) (ext.∠ of ∆)
\(\begin{array}{c}\angle BCD + \angle DEF = 180^\circ \\x + 65^\circ = 180^\circ \\x = \underline{\underline {115^\circ }} \end{array}\) (opp.∠, cyclic quad.)

Theory
an exterior angle of a cyclic quadrilateral equals its interior opposite angle

ExamplesIn the figure, chords AD and BC are produced to meet at E. If \(\angle BAE = 60^\circ \)
and \(\angle AEB = 40^\circ \), find x. 
Solutions\(\begin{array}{1}\angle DCE = \angle BAD\\ = 60{^\circ {}}\end{array}\) (ext. , cyclic quad.)
In CDE,
\(\begin{array}{1}x = \angle DCE + \angle CED\\ = 60^\circ + 40{^\circ {}}\\ = {\underline{\underline {100^\circ }} {}}\end{array}\) (ext. of )