Geometry question?

Let's name the points first. Let A be the upper point, B left low, C right low, D left middle and E right middle.

ADC =180-BAC-ACD=180-20-20=140

BDC = 180-140=40

DCB = 180-DBC-BDC=180-(10+70)-40=60

BEC = 180-EBC-BCD-DCE=180-70-60-20=30

sin(BEC)/BC = sin(ECB)/BE <=> BC = BE*sin(BEC)/sin(ECB)

sin(BDC)/BC = sin(DCB)/BD <=> BC = BD*sin(BDC)/sin(DCB)

BD*sin(BDC)/sin(DCB) = BE*sin(BEC)/sin(ECB)

BE/BD = sin(BDC) / sin(DCB) / (sin(BEC) / sin(ECB))

BE/BD = sin(40) / sin(60) / (sin(30) / sin(80))

BE/BD = 1.461902... (sorry, just temporary)

DE = sqrt(BD^2 + BE^2 - 2*BD*BE*cos(DBE))

DE = BD*sqrt(1 + (BE/BD)^2 - 2*BE/BD*cos(DBE))

DE/BD = sqrt(1 + (BE/BD)^2 - 2*BE/BD*cos(DBE))

BD/DE = 1/sqrt(1 + (BE/BD)^2 - 2*BE/BD*cos(DBE))

BD/DE = 1/sqrt(1 + 1.461902^2 - 2*1.461902*cos(10)) = 1.969616...

sin(x)/BD = sin(DBE) / DE

sin(x) = BD / DE * sin(DBE)

sin(x) = 1.969616 * sin(10) = 0.34202

x = 20

I'm sure this is not the agreed solution, it is obtained through brute force but at least it gives us a hint. It is like consulting the answer at the end of the book.