Question

The question in the comments

Answer 1

my first approach would be to find all 4 angles within ABCD as a function of x, the let the sum be 360

Answer 2

we know A is 3x

Answer 3

B has to be 180-5x and D has to be 180-4x

Answer 4

now we need to know what is DCB, well , its just Q+QDC

Answer 5

but QDC is 4x, so Q+QDC is 6x and DCB is 6x

Answer 6

shoot, looks like I got nothing

Answer 7

Hints:
1. In cyclic quadrilaterals, opposite angles are supplementary.
2. exterior angle of a triangle=sum of non-adjacent angles
3. adjacent angles on a line are supplementary

Answer 8

DC + CB == 6 x
BA - DC == 4 x
AD - CB == 2 x

AD + BA == 360 - 6 x

You find

BA= 240 - AD,
CB = -40 + AD,
DC = 160 - AD
x=20

For example if we take AD=80, we get
BA=160
CB=40
DC=80

Answer 9

All the BA, CB .... are arcs meauserments

Answer 10

As you can see,
angle B+angle D=180 degrees which is expected of a cyclic quadrilateral, so
x=20 is correct!
Well done! :)