SOLUTION: ABDE is a square section and BDC is an equilateral triangle with C outside the square. Compute the value of angle ACE.

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Question 896311: ABDE is a square section and BDC is an equilateral triangle with C outside the square. Compute the value of angle ACE.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!

see the following diagram.
further comments are below the diagram.

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your equilateral triangle is BCD.
since it's an equilateral triangle, all angles are equal to 60 degrees.
you draw a line from A to C and from E to C.
this forms 2 isosceles triangles of ABC and EDC.
the congruent legs of triangle ABC are:
AB and BC
the congruent legs of triangle EDC are:
ED and DC
the angles opposite these congruent legs are equal.
the angles that are equal for triangle ABC are:
angle BAC and angle BCA
the angles that are equal for triangle EDC are:
angle DEC and angle DCE
angle ABC is composed of angle ABD which is 90 degrees and angle CBD which is 60 degrees. angle ABC is therefore equal to 150 degrees because 90 + 60 = 150.
since the sum of the angles of a triangle is equal to 180 degrees, this means that angles BAC and BCA must be equal to 30 degrees in total because angle ABC is equal to 150 degrees.
since those angles are equal, then each of them must be equal to 15 degrees.
similarly angles DEC and DCE must each be equal to 15 degrees also.
the angle in question is angle ACE which is the third part of angle C which is equal to 60 degrees.
the first part is angle BCA which is equal to 15 degrees.
the second part is angle DCE which is equal to 15 degrees.
the third part is angle ACE which has to be equal to 30 degrees since 60 - 15 - 15 = 30.

that's your answer.
angle ACE is equal to 30 degrees.