Question

What is the range of possible values for x?

Answer 1

The minimum value for 2x is 0 
the maximum value is achieved when A, D and C are collinear and the quadrilateral ABCD becomes an isosceles triangle ABC 
base AB = 52 and vertical angle 2x + 34° 

For the sine law 
(sin 2x)/22 = (sin ADB)/AB 
(sin 34°)/30 = (sin BDC)/BC 

is given that AB = BC, and sin ADC = sin BDC because they are supplementary, so from 
(sin ADC)/AB = (sin BDC)/BC 

it follows 
(sin 2x)/22 = (sin 34°)/30 

sin 2x = 22 (sin 34°)/30 

2x = asin(22 (sin 34°)/30) ≈ 24.2° 

x = 0.5 asin(22 (sin 34°)/30) ≈ 12.1° 

0 < x < 12.1°