let us start with the namings first this is the center of a circle where all the points on the circle are at an equal distance from this point now if we draw a line from the center to any point on the circle this line is called the radius the radius is the same for all points on the circle if we extend a straight line across the circle passing through the center we get the diameter the diameter is always twice the length of the radius a line segment that connects any two points on the circle without necessarily passing through the center is called a cord the longest possible chord of a circle is its diameter a curved part of the circle between two points is known as an arc while a sector is the region or an area between two radi and an arc if a chord divides the circle into two regions each region is called a segment now the circumference of a circle is the total length of its boundary or or perimeter of a circle then a tangent to a circle is a straight line that touches the circle at exactly one point without Crossing it and a secant is a straight line that intersects The Circle at two distinct points then we use the word subtended to describe how an arc or a chord creates an angle at a point on the circle for example when an arc forms an angle at the circumference we say the angle subtended at the circumference by The Arc similarly if the angle is formed at the center we say the angle subtended at the center by The Arc this completes the basic naming of a circle now theorem number one angles subtended by the same Arc are equal this means that if an arc subtends an angle at multiple points on the circle all such angles are equal so this angle is equal to this and this one as well now if I draw another angle like this one it will be equal to all of these angles now here's a question if this angle is 65° and this angle is 35° then what will be the value of all of the remaining angles it's easy right this is 65° so this angle will also be equal to 65° because angles subtended by the same Arc are always equal similarly if this angle is 35° then this will also be equal to 35° following the same theorem now the sum of all angles in a triangle is 180° so can you tell me in the comments what will be this angle let us move on to theorem number two angle at the center is twice the angle at the circumference which means the angle subtended by an arc at the center of the circle or this angle is always twice the angle subtended at the circumference by the same Arc so if this angle is 40° then this central angle will be twice of it or 80° but if I draw like this then this angle will be 40° using the first theorem we just saw now theorem number three angle in a semicircle is always 90° which means if an arc is a semicircle that is its end points lie on a diameter then the angle subtended at the circumference is always always 90° using theorem 2 can you tell me why is this so this is because we know that a full circle has 360° so half of a circle has 180° therefore this angle which is subtended by this semicircle on the circumference will be half of it or 90° right noise okay tell me if this is the diameter of the circle this side length is three and this is four then what will be the value of the diameter now theorem 4 theorem of cyclic quadrilateral a quadrilateral is called cyclic if all its four vertices lie on a circle the important property of a cyclic quadrilateral is that its opposite angles are always supplementary this means that if we take any two opposite angles of a cyclic quadrilateral and add them together the result will always be 180° for example if this angle is 70° then this angle will be equal to 180- 70 or 110° now theorem 5 theorem of tangent and radius we have already mentioned that a tangent is a straight line that touches the circle at exactly one point without Crossing it the important property of a tangent is that it is always perpendicular to the radius at the point where it touches the circle now theorem six the two tangents theorem if two tangents are drawn to a circle from a single external point then these tangents will always be equal in length another important property of this theorem is that the line joining the external point to the center of the circle always bcts the angle formed between the two tangents so if this angle is 40° then both these angles will be equal to 20° and if this length is equal to 5 units then this piece will also be equal to 5 units now theorem 7 alternate segment theorem this theorem states that the angle between a tangent and a chord drawn from the point of contact is always equal to the angle that the same chord subtends in the opposite segment of the circle same is true for this angle it will be equal to this angle because this chord makes this angle at the circumference of the circle now theorem 8 chord bis sector theorem this means that if we draw a line that is both perpendicular to a chord and divides it into two equal parts then that line must pass through the center of the circle now theorem 9 equal chords equal angles theorem suppose we have a circle with Center o and two chords ab and C D of equal length if we draw lines from o to the end points of both chords then the angles a o b and c o d at the center will be equal similarly if we take any point p on the circumference and connect it to A and B to form angle APB and another Point Q connected to C and D to form angle cqd then these two angles will also be equal now theorem 10 2 second segment theorem suppose we have a circle and an external Point P outside the circle from P we draw a second line that intersects The Circle at two points A and B and we draw another second line that intersects The Circle at two points C and D then according to the theorem the product of the distances PA * PB will be the same as PC * PD awesome we can extend this theorem and say that if this line is a tangent say this line touches the circle at a single point C then we have PA * PB = P c² this theorem is called the tangent secant theorem which is very similar to two secant segment theorem now suppose we have a circle and an external Point P outside the circle from P we draw a secant line that intersects The Circle at two points A and B such that this secant line passes through the center of this circle then we draw this tangent which touches the circle at C now if PA is three units and PC is four units then what will be the radius of this circle we will solve it using two different methods first we will use tangent secant theorem let us label the radius as r so AB or the diameter of this circle will be 2 R thus PB will be equal to 3 + 2 R right now using this theorem we get 4² = 3 * 3 + 2 r or 16 = 9 + 6 R therefore R = 16 - 9 or 7 / 6 we can also solve the same using the fact that a tangent is always perpendicular to the radius at the point where it touches the circle so this will form a right angled triangle with the side lengths as 4 R and the hypotenuse as 3 + r and thus 3 + r sare = 4² + r² expand it to get this r² gets canceled out and we are left with 6 r + 9 = 16 or r = 7 / 6 Now we move on to our final theorem or theorem 11 which is Chord chord power theorem this theorem states that if two chords intersect inside a circle let us label the sides as a b c and d the theorem says a * B will always be equal C * D let us end this video with this question suppose this is a cyclic quadrilateral in which these are its diagonals if this angle is 55° and this angle is 45° then what will be the value of this angle okay using our first theorem we know that the angles subtended by the same Arc are equal so if this angle is 55° then this angle will also be equal to 55° so this angle will be equal to 55 + 45 or 100° then we also know that the opposite angles of a cyclic quadrilateral are supplementary that is they add to 180° so if this is 100 then this will be 80° and that's it aren't circles cool if you enjoy my videos and want to support my channel consider becoming a patreon as it helps me create more awesome content for you link is in the description also you can support my Channel by joining our community and becoming a member so good now click on this video to solve a very nice Circle problem which will use many of the theorems discussed here