# In quadrilateral $ABCD$, $AD$ is parallel to $BC$ and $AB= AC$ . $F$ is a point on $BC$ such that $DF$ is perpendicular to $BC$. find $∠BAD$ .

In quadrilateral $$ABCD$$, $$AD$$ is parallel to $$BC$$ and $$AB= AC$$ . $$F$$ is a point on $$BC$$ such that $$DF$$ is perpendicular to $$BC$$. $$AC$$ intersects $$DF$$ at $$E$$. If $$BE= 2DF$$ and $$BE$$ bisects $$\angle ABC$$, find the measure, in degrees, of $$∠BAD$$ . I worked out that $$\angle ABC = \angle ACB =\angle DAE$$

I think the angle bisector theorem could be used but im not sure how to use it in a way that would incorporate the fact that $$BE=2DF$$

Hint. Consider the perpendicular to $$BC$$ at $$B$$ and let $$G$$ be its intersection with line $$AC$$. Since triangle $$ABC$$ is isosceles, we have that its height relative to side $$BC$$ intersects $$BC$$ in its midpoint. Thus, $$BG = 2DF$$ and triangle $$BGE$$ is isosceles.
• don't you mean $BGA$ is isosceles, I don't understand how $BGE$ is isosceles? Aug 27 '19 at 18:33
• We are using that $BG = 2DF = BE$, where the second equality is a hypothesis. Aug 27 '19 at 18:36