I want to find $\angle BAD$.

Question

$ \angle BDC = 120°$, $\angle BEC = 160°$. I want to find $\angle BAD$.

Answer 1

Let $\alpha = \angle BAD = \angle CAD$, $\beta = \angle ABD = \angle EBD$, $\gamma = \angle ACD = \angle ECD$.

The total of the angles in the quadrilateral $BACE\,$ is $$ 2\alpha + 2\beta + 2\gamma + 200^\circ = 360^\circ, $$ so $$ \alpha + \beta + \gamma = 80^\circ. $$ The total of the angles in quadrilateral $BDCE\,$ is $$ \beta + 120^\circ + \gamma + 200^\circ = 360^\circ, $$ so $$ \beta + \gamma = 40^\circ. $$ Subtracting this from the previous result, $\alpha = 40^\circ$.