Cyclic Quadrilateral in Circle Geometry

A cyclic quadrilateral is inscribed into a circle, whose vertices all lie on a circle. The properties of cyclic quadrilateral in circle geometry are;

  1. Opposite angles in a cyclic quadrilateral supplementary.
  2. Exterior angle and its opposite angle are equal.

(a)    Prove that \( FADG \) is a cyclic quadrilateral.

\( \begin{aligned} \require{color}
\text{Let } \angle BCD &=\theta \\
\angle FAD &= \theta &\color{red} \text{opposite angle’s of a cyclic quadrilateral} \\
\angle FGD &= 180^{\circ} – \theta &\color{red} \text{co-interior angles are supplementary} \\
\angle FAD + \angle FGD &= 180^{\circ} \\
\therefore FADG \text{ is a } &\text{cyclic quadrilateral} &\color{red} \text{opposite angles are supplementary} \\
\end{aligned} \)

(b)    Prove that \( GA \) is a tangent to the circle through \(A\), \(B\), \(C\) and \(D\).

\( \begin{aligned} \require{color}
\angle DAG &= \angle DFG &\color{red} \text{angles in the same segment} \\
\angle DFG &= \angle AEF &\color{red} \text{alternate angles in parallel lines} \\
\angle DAG &= \angle AED \\
\therefore GA \text{ is a } &\text{tangent} &\color{red} \text{alternate segment} \\
\end{aligned} \)

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