9/04/2014 Consider regular pentagon ABCDE such that DEFG is a square. Find the measures of ∠EAF, ∠AFD and ∠DAF.

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9/04/2014 Solution. Sketch a figure to satisfy the given conditions.

Since ABCDE is a regular pentagon, the interior angle ∠AED = 108°, and ∆ADE is an isosceles triangle, ∠ADE = 36°. Also, ∠AEF = 162° since ∠DEF = 90° and ∠AED = 108° [subtracting from 360°]. As ∆AEF is an isosceles triangle, ∠EAF = 9° = ∠AFE. Therefore, ∠DAF = 36° + 9° = 45° and ∠AFD = 45° + 9° = 54°.

∠EAF = 9°, ∠AFD = 54° and ∠DAF = 45°.