Circle Theorem - Alternate Segment Question Hi there I have a maths question from my GCSE book which is just really bewildering me and my teacher. I have taken the maths question out from my book and made a computerized version and this is what I got. Please bare in mind that the diagram is not fully accurate and is supposed to be able to be solved but I can't find anything that will help me find the value of S. I know this site is for advanced questions but could someone please tell me how to solve this question. I know that the answer to it is 62° (As it says in the book) but could someone please show me how they got the answer as I am trying all 9 of the circle theorem rules that I have come across and none of them could help me find the logical explanation of answering this question. Please also bare in mind that this is a GCSE A* type question and I don't think it should need anything too advanced
Thanks

 A: As it stands, your figure is ambiguous.
In the figure below, any point on the (big) arc $\stackrel{\frown}{ST}$ is a point $P$ such that $m\angle SPT = 62^\circ$, and each one of those points makes a differently-sized angle $\angle PSU$.

Unless you can provide more information, the question has no single answer.

*

*@A1D1S's answer assume that $\triangle SPT$ is isosceles with vertex $P$, but the resulting measure ($59^\circ$) for $\angle PSU$ didn't match your book. (Books can be wrong, of course.)


*If it happens that $\triangle SPT$ is isosceles with vertex $S$, then $m\angle PSU$ will be $62^\circ$ for the same reason that $m\angle TSR = 62^\circ$. This matches your book. (Books can be right, too.)


*If $\triangle SPT$ is isosceles with vertex $T$, then $m\angle PSU = 56^\circ$.


*Scenarios exist that justify $m\angle PSU$ being anything from $0^\circ$ to $118^\circ$.
A: Assume an arbitrary center, consider the angle subtended by the arc (that subtends the angle 62) at the center. It will be 124 degrees.
Now tangents are perpendicular to the radii so 124 + 90 +90 + x = 360
x = 56. This is the value of the angle between the tangents.
and since the length of the tangents are the same length, isosceles triangle, so 56 + 2y = 180 ; y = 62.
similarly on the inside triangle isosceles condition if used gives 62 + 2z = 180; z= 59
finally 62 + 59 + S = 180 ; S = 59
