### Basic properties of circles - (10.3)

#### Understand the properties of a cyclic quadrilateral

• the opposite angles of a cyclic quadrilateral are supplementary

• In 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.
• In∆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.)

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

• In the figure, chords AD and BC are produced to meet at E. If $$\angle BAE = 60^\circ$$
and $$\angle AEB = 40^\circ$$, find x.
• $$\begin{array}{1}\angle DCE = \angle BAD\\ = 60{^\circ {}}\end{array}$$ (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 )