 Hello and welcome to the session. In this session first we will discuss angles subtended by an arc of a circle. We have a fact that if two quads of a circle are equal then their corresponding arcs are congruent and conversely if two arcs are congruent, corresponding quads are equal. Like if you consider the quads AB and PQ and suppose that we have quad AB is equal to the quad PQ then the corresponding arcs are congruent that is the arc AB is congruent to the arc PQ. And if in case we have that arc AB is congruent to the arc PQ then this gives us that the quad AB is equal to the quad PQ. Now we define angle subtended by an arc at the centre is defined to be the angle subtended by the corresponding quad at the centre. Therefore we have angle subtended by the minor arc PQ is the angle POQ. Now the angle subtended by the minor arc PQ that is this is the angle POQ and the angle subtended by the major arc PQ is the reflex angle POQ that is this angle. We have a following result according to which we have congruent arcs or we can say equal arcs of a circle subtend equal angles at the centre. Now the relationship between the angles subtended by an arc at the centre and a point on the circle is given by the result which says 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 that is we have angle subtended by the arc PQ at the centre is angle POQ and this is the double that is two times the angle subtended by the minor arc PQ at any point on the remaining part of the circle at this point be the point R. So this would be two times the angle PRQ. Suppose that we join the points P and Q so we get a cot PQ then this angle PRQ is the angle formed in the segment PRQP that is this segment and we have a result related to the angles in the segment which says that angles in the same segment of a circle are equal. So like in this case we get that angle PRQ that is this angle is equal to the angle PSQ that is these are the two angles in the same segment. Now the converse of this statement is given by the result which says that if a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment the four points lie on a circle that is they are consyclic like in this case we have that the line segment joining the points P and Q subtends equal angles at the points R and S which lie on the same side of the line containing the line segment PQ that is we have angle PRQ is equal to the angle PSQ then we say that the four points lie on a circle that is the points PQ R and S lie on a circle or we can say that PQ R is a consyclic we have one more result which says that angle in a semicircle is a right angle like this angle BCA is an angle in a semicircle so angle BCA is equal to 90 degrees next we shall discuss cyclic quadrilaterals a quadrilateral is called cyclic if all the four vertices of it lie on the circle like in this figure as you can see that the quadrilateral ABCD has all its four vertices A, B, C and D on the circle so ABCD is a cyclic quadrilateral now we have a result related to cyclic quadrilateral which says that the sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees like for this cyclic quadrilateral ABCD we have angle A plus angle C is equal to 180 degrees or angle B plus angle D is 180 degrees and the converse of this result is given by the statement if the sum of a pair of opposite angles of a quadrilateral is 180 degrees the quadrilateral is cyclic like for this figure if we are given that angle A plus angle C is 180 degrees or we are given angle B plus angle D is equal to 180 degrees then we can conclude that ABCD is a cyclic quadrilateral in this given figure we have O is the center of the circle an angle ADC is equal to 130 degrees and we have angle BAC is equal to X we need to find the value for X so as you can see ABCD is a cyclic quadrilateral since all its four vertices lie on the circle thus we have angle ADC plus angle ABC is equal to 180 degrees since we know a pair of opposite angle of a cyclic quadrilateral is 180 degrees so from here we get angle ABC is equal to 180 degrees minus angle ADC which is 130 degrees and this is equal to 50 degrees then we have angle ACB would be equal to 90 degrees since it is the angle in a semicircle now let's consider triangle ABC in this we have angle ADC plus angle ACB plus angle BAC is equal to 180 degrees by the angles and property of a triangle let's substitute the value for angle ADC which is 50 degrees plus angle ACB that is 90 degrees plus angle BAC that is X is equal to 180 degrees so from here we get X equal to 180 degrees minus 140 degrees this gives us X equal to 40 degrees this completes the session hope you have understood the concept of angles subtended by an arc of a circle and cyclic quadrilaterals