In Geometry (Class 9 - Circles), combining quadrilaterals (four-sided shapes) with circles creates a special shape known as a Cyclic Quadrilateral. Let's look at its defining properties.
Definition: All 4 corners must touch the circle's boundary.
Theorem 1: Opposite angles sum up to 180 degrees.
Theorem 2: Exterior angle equals the opposite interior angle.
A quadrilateral is called a Cyclic Quadrilateral if all four of its vertices (corners) touch the circumference of a single circle. If even one corner is inside or outside the circle, it is NOT a cyclic quadrilateral.
The most important theorem you will use to solve problems is regarding its angles: The sum of either pair of opposite angles of a cyclic quadrilateral is exactly 180° (They are supplementary).
For example, if ABCD is a cyclic quadrilateral:
(Note: The reverse is also true. If the opposite angles of any quadrilateral add up to 180°, that quadrilateral is guaranteed to be cyclic).
If you extend one side of a cyclic quadrilateral outwards, it forms an exterior angle. The exterior angle of a cyclic quadrilateral is always equal to the interior opposite angle.
The main property of a cyclic quadrilateral is that the sum of its opposite interior angles is always 180 degrees. Also, if one side is extended, the exterior angle formed is equal to the opposite interior angle.
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