# Common rules in calculating angles of diagonal lines in quadrangle, using synthetic(axiomatic) geometry. In this quadrangle, if $$x, y, a, b$$ is determined,

$$c$$ has to be determined too.

Can $$c$$ be expressed using $$x, y, a, b$$?

Searching similar questions in axiomatic geometry is hard.

If similar question is present, please let me know.

• What have you tried? Hint: there are several triangles whose angles are all known.
– dxiv
Aug 22, 2021 at 1:59
• @dxiv I have tried making a triangle EBC (E is a new dot that makes EBC a triangle, which is at the upper side of ABCD) Can you be more specific about the hint?
– SGKw
Aug 22, 2021 at 7:00
• Let $\,BC=d\,$, then you can calculate all sides and diagonals in terms of $\,d\,$ and the given angles, then you can solve one of the triangles for the $\,c\,$ angle.
– dxiv
Aug 22, 2021 at 7:17
• @dxiv I assume letting $BC=d$ is not pure axiomatic geometry...? If you could hint or answer with drawings, that would be grateful. Thank you.
– SGKw
Aug 23, 2021 at 0:13
• @dxiv Thank you for your explanation, now I know that only in special circumstances, $c$ can be calculated without using the law of sines/cosines and trig functions. I posted a new question related to this math.stackexchange.com/questions/4230763/…
– SGKw
Aug 23, 2021 at 2:36