Page 1 - Lesson note 5 Angle subtended by arc Ch.- 10 Circle

SAI International School

Class-IX

Mathematics
Chapter-10: Circle

Lesson Note:-6

Sub topic : Angle Subtended by an arc of circle.

We have seen that the end points of a chord other than diameter of a circle cut it into two arcs –
one major and other minor.
If we take two equal chords then corresponding arcs equal in length.
They are congruent in the sense that if one arc is put on the other, without bending or twisting, one
superimposes the other completely.
If two chords of a circle are equal, then their corresponding arcs are congruent and
conversely, if two arcs are congruent, then their corresponding chords are equal
The angle subtended by an arc at the centre is defined to be angle subtended by the
corresponding chord at the centre in the sense that the minor arc subtends the angle and the
major arc subtends the reflex angle. In the figure the angle subtended by the minor arc PQ at O is
POQ and the angle subtended by the major arc PQ at O is reflex POQ.

Congruent arcs (or equal arcs) of a circle subtend equal angles at the centre.

Therefore, the angle subtended by a chord of a circle at its centre is equal to the
angle subtended by the corresponding (minor) arc at the centre.

Theorem 10.8 : The angle subtended by an arc at the centre is double the angle
Subtended by it at any point on the remaining part of the circle.