I will go ahead and admit, this might just be something obvious but I did research and couldn't find anything.

I have a pentagon, and I know two top points (A & B) and the distance between them (black). I also know the other 4 side lengths (blue), which are all the same. I know the bottom point too, and I don't care about the two side points.

How can I use the bottom point (E) to determine the angle between each of the top two points, and their adjacent side lengths?


enter image description here

Points A, B, & E are known. All side lengths are known. Side lengths of the same colour are equal. Angles that I want to know have dotted lines.

EDIT II: I realize my question may need clarification. The pentagon is on a coordinate plane. I know the location of A, B, and E. I want to use point E to find the interior angles on points A & B.

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  • $\begingroup$ It’s difficult to see exactly what you’re asking, could you draw a picture labeled with the information you know and the angle you would like to determine? $\endgroup$ – Alex Jan 26 '19 at 13:28
  • $\begingroup$ @Alex Thank you. I have added a picture and some clarification $\endgroup$ – 12buntu Jan 26 '19 at 14:52

You want to find angle $\alpha+\beta$ in diagram below, and the analogous angle of vertex $B$ (caution: names of points are different from those in the question). By standard trigonometry we have: $$ \cos\alpha={AH\over AD}={{1\over2}AC\over AD}, \quad \cos\beta={AB^2+AC^2-BC^2\over2\,AB\cdot AC}. $$ And similar formulas for the angles of vertex $B$.

enter image description here

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  • $\begingroup$ Sorry that my question wasn't clear. I have since edited it and included a picture for clarification. $\endgroup$ – 12buntu Jan 26 '19 at 14:44
  • $\begingroup$ My solution works fine even if $AB$ is different from the other sides. $\endgroup$ – Aretino Jan 26 '19 at 14:55

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