The value of x in the diagram is 35°
Secant-secant angles are formed between two secant lines intersecting outside a circle.
The measure of these angles is half the difference of the intercepted arcs along the circle.
Other theorem in circle geometry include: Arc substended at the centre is equal to the measure of the arc substended by it.
Therefore ;
137 = 1/2( 309 - x)
309 - x = 274
x = 309 - 274
x = 35°
Therefore, the value of x in the diagram is 35°