Angle Properties of Circle

1. Angles at Centre and Angles at Circumference
   
   \(\angle AOB\) and \(\angle COD\) are angles subtended at the centre of the circle by the minor arc
   \(\angle AXB\) and \(\angle CXD\) are angles subtended at the circumference of the circle by the same minor arc
   
   \(\angle AOB\) and \(\angle COD\) are angles subtended at the centre of the circle by the major arc
   \(\angle AYB\) and \(\angle CYD\) are angles subtended at the circumference of the circle by the same major arc
2. Circle Angle Property \(1\)
   An angle at the centre of the circle is twice that of any angle at the circumference subtended by the same arc.
   
   \(\angle AOB = 2\angle APB\) (angle at centre \(= 2\) angle at circumference)
   ** Property involve centre of circle.
3. Circle Angle Property \(2\)
   An angle in a semicircle is always equal to \(90{}^\circ\).
   
   \(\angle ACB = 90{}^\circ\) (angle in a semicircle)
   ** Property involve diameter of circle
4. Circle Angle Property \(3\)
   Angles in the same segment are equal
   
   \(\angle ADB = \angle ACB\) (angles in the same segment are equal)
   ** Property involve all \(4\) points lying on the circumference of circle.
5. Circle Angle Property \(4\)
   Angles in the opposite segments are supplementary, i.e. they add up to \(180{}^\circ\).
   
   \(\angle ABC + \angle ADC = 180{}^\circ\), \(\angle DAB + \angle DCB = 180{}^\circ\) (angles in opposite segments are supplementary)
   ** Property involve all \(4\) points lying on the circumference of circle.

For the same-segment equality, the two angle vertices lie on the same side of the chord. On opposite sides, the angles are supplementary. Use the reflex central angle when the corresponding arc is the major arc.