21. Angle subtended at the centre The angle subtended at the centre from an arc is double the angle at the circumference.yx180 – 2xCxB180 – 2y2x+2y= 2(x+y)yA
32. Angles subtended from the same arc. Angles subtended from the same arc are equal.CBA
43. Angles in a semi-circle. The largest angle in a semi-circle will always be 9090CAB
54. The Angle between a Tangent and its radius. Definition:A tangent is a line that will touch the circle at one point only. (i.e. it does not cut the circle)CTangentAssume that the tangent is not perpendicular to the radius. Now there must be a perpendicular from the center of the circle to the tangent. It must intersect the tangent elsewhere (apart from the point on intersection). Now since a perpendicular is the shortest distance of a point from a line, the perpendicular must have distance < radius of the circle (since line from point of intersection of tangent to center of circle is a radius). => the foot of the perpendicular must lie within the circle. =>the line meets the circle elsewhere and the line is not really a tangent! Thus there is a contradiction. Therefore the radius is the shortest distance. Consequently, it must be perpendicular to the tangent at the point of intersection90AThe angle between a tangent an its radius will always be 90
65. Angles in a cyclic quadrilateral. Definition:A cyclic quadrilateral is any four-sided polygon whose four corners touch the circumference of the circle180 – db2360 –2Opposite angles in a cyclic quadrilateral add up to 180c
74. The Angle between a Tangent and a chord. Definition: A chord is any straight line which touches the circumference at two points. The largest chord possible is called the diameter.=180 – 90 – (90 – )=Tangent9090 –ChordThe angle between a tangent a chord is equal to the angle in the alternate segment.