8.1 Complete the following: The opposite angles of a cyclic quadrilateral are … 8.2 In the diagram, EF and EG are tangents to circle with centre O - NSC Mathematics - Question 8 - 2016 - Paper 2

To prove that FOGE is a cyclic quadrilateral, we must show that the opposite angles are supplementary. Since angles F and G subtend arc OE, we have:

$F + G = 180°$

Thus, FOGE is a cyclic quadrilateral.

EG is a tangent to circle GJK because it touches the circle at point G. By the tangent-secant theorem, the angle formed between the tangent (EG) and the secant (JK) meets the radius (OG) at point G, confirming EG's tangency.

To find the angle FEG, we note that it is equal to the angle subtended by the arc FK. Using the tangent properties and the relationship between angles and arcs, we can express it as:

$ext{Angle FEG = 180° - 2x}$

This follows from the properties of cyclic quadrilaterals and the inscribed angle theorem.