 Hello and welcome to the session let us discuss the following question. It says prove that if a line touches a circle and from the point of contact a chord is drawn the angles which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments. Using the above do the following. AB is a diameter and AC is a chord of the circle so that angle BAC is 30 degrees. The tangent at C intersects AB produced in a point D prove that BC is equal to BT. Let's now move on to the solution and let us first write what is given to us. We are given that Pq is a tangent to a circle with center O at a point A. Two segments of the circle by the chord AB have to prove that the angles which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments. Now as we have to prove Aq that is this angle equal to the angle ACB and also angle BAP equal to angle let's now do some construction we draw a diameter so we have drawn a diameter AOA and we have joint AB and what we have to prove is angle BAq is equal to angle ACB that is this angle is equal to this angle and we have to prove that angle BAP that is this angle is equal to angle ADP that is this angle. So let's now start the proof 90 degrees because it's an angle in semicircle therefore angle AEB AB that is this angle plus this angle is 90 degrees the angles of the triangle is 180 degrees since this angle is 90 degrees so some of these two angles will be 90 degrees. Now again it's perpendicular to Pq that is this angle 90 degrees that is angle EAB plus angle BAq is 90 degrees and angle EAB plus angle BAq is angle EAq is perpendicular to this is because diameter and Pq is the tangent and diameter is always perpendicular to a tangent to the circle. Let us name this as 1 and this as 2 now from 1 and 2 angle AEB equal to angle BAq and we can see that angle AEB plus angle EAB is 90 degrees and there also angle EAB plus angle BAq is 90 degrees so from this we can say that angle AEB is equal to angle BAq also ACB angle AEB plus these are angles in same segment angle AEB is equal to angle BAq and also angle AEB is equal to angle ACB so we have angle ACB is equal to angle BAq so we have proved that angle ACB is equal to angle BAq now again BAq plus angle BAp is 180 degrees because it's a linear pair 180 degrees because these are the opposite angles of a cycle quadrilateral these two we can see that angle BAq plus angle BAp is equal to angle ACB plus angle ADB now from here let us name this as 3 below that angle BAq is equal to angle ACB so this gets cancelled from both sides till BAq is equal to angle ACB BAp is equal to angle ADB and this is what we had to prove hence we have proved that angle BAq is equal to angle ACB and angle BAp is equal to angle ADB now the next part we have to prove that AB is the diameter and AC is the chord of a circle so that angle BAc is 30 degrees that is this angle is 30 degrees the engine that see intersect AB produced in D we have to prove that BC is equal to VT so let's now move on to the second part now we are given that AB is the diameter but angle BAC is equal to 30 degrees this is given to us we have to prove BC is equal to VT is 90 degrees because it's angle is semicircle let's name this as angle 1 this as 2 this as 3 this as 4 and this as 5 now in triangle ABC for angle 5 is equal to angle 1 plus angle 2 angle 1 is 30 degrees angle 2 is 90 degrees so angle 1 angle 5 is 30 plus 90 degrees that is 120 degrees now also angle 1 is equal to angle 3 because these are the angles in alternate segments by the theorem we just proved angles in alternate segment for angle 3 is equal to 30 degrees we have to prove that BC is equal to BD for that we'll prove that angle 3 is equal to angle 4 this is by the property of isosceles triangle which says that if the opposite angles are equal then sides will be equal that is sides opposite the equal angles are equal so we have to prove that angle 4 is also of 30 degrees but angle 3 plus angle 4 plus angle 5 is 180 degrees as we know that sum of three angles in a triangle is 180 degrees and this angle sum property of triangle the 3 is 30 degrees plus angle 4 angle 5 is 120 degrees is equal to 180 degrees so angle 4 is equal to 180 degrees minus 20 degrees minus 30 degrees so this is equal to 180 degrees minus 150 degrees that is 30 degrees so we have got that angle 3 is equal to angle 4 so this implies BC is equal to BD that is sides opposite the equal angles is what we had to prove so there is the result is proved so this completes the question and the session by for now take care how good they