Basic properties of circles - (10.3)

Understand the properties of a cyclic quadrilateral

  • Theory


    the opposite angles of a cyclic quadrilateral are supplementary
  • Examples
    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.
  • Solutions

    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.)

  • Theory


    an exterior angle of a cyclic quadrilateral equals its interior opposite angle
  • Examples
    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.
  • 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 )