 Hello everyone. How are you all today? The question says if a line touches a circle and from the point of contact a chord is drawn Prove that the angles which the chord make with the given line are equal respectively to the angles Formed in the corresponding alternate segment. Now using the above theorem we need to prove the following P is the midpoint of arc A, B A, B, B prove that tangent q are drawn at P to the circle is parallel to A, B Right, so first of all We need to prove the first part for this question that this is a required figure Which we need here. We will be writing what is given to us Now here we are given that pq is a tangent to the circle with center o and radius r with point of contact a A, B is a chord and D and C are points on the arc A, B and arc B are respectively other than A and B We need to prove that angle B, A, q, B, A, q is equal to A, C, B, A, B is equal to A, D, B For that we need to draw the diameter A, O, E and join B, E. Now, let's proceed with the proof In triangle A, B, E We know that A, B, E is equal to 90 degrees because it is an angle in a semi Circle as we have by construction drawn A, O, E as diameter Now further angle A, E, B is plus angle A, B, L, V, A, E is equal to 180 degrees Because 180 degrees, 90 degrees and the sum of the remaining two angles Become equal to 90 degrees itself. So angle A, E, B plus angle B, A, q becomes equal to 90 degrees Let this be the first. We can see that equal to 90 degrees. Let this be the second equation and this is because It is the angle between the tangent, the second equation we get. Let this be the third equation that to be done over to 180 degrees. Similarly, angle B, this is because of linear pair They are forming linear pair and this is because the correlator B, A, q because they both are equal to 180 degrees So hence their sum will be equal to each other from the fifth and the eighth equation. Now here This was the eighth equation and the above were the sixth and seventh We get angle A, angle A, D, B to angle P, A, B plus angle A, C, B or we can say that First we can say that angle P, A, B is equal to angle A, D, B and this is the required Proof of the theorem which was given to us. This completes the first part of the solution Now let's proceed with the second part Here we are given that T is the midpoint of R for convenience. Name these four rounds Begin with our proof. We can conclude that A, P is A. The whole question well, so have a nice day