Question 997740: Good day! Can you please help me with my homework,
A square section ABCD has one of its sides equal to x. Point E is inside the square forming an equilateral triangle BEC with one side equal in length to the side of the square. Find angle AED.
Thank you!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your square is ABCD
you have a point within the square that forms an equilateral triangle BEC.
the 3 sides of the equilateral triangle are sides EB, EC, and BC.
since BC is one of the sides of the triangle and is also one of the sides of the square, then each side of the equilateral triangle must be congruent to each side of the square.
since BEC is an equilateral triangle, then each of its angles is equal to 60 degrees.
what you get is:
angle BEC = 60 degrees.
angle EBC = 60 degrees.
angle ECB = 60 degrees.
since each angle of a square is equal to 90 degrees, and one part of the angle is 60 degrees, then the other part of the angle must be 30 degrees because 60 + 30 = 90.
you get:
angle ABE = 30 degrees
angle DCE = 30 degrees.
2 isosceles triangle are formed from the congruent sides.
those 2 isosceles triangle are:
triangle ABE and triangle DCE.
since their vertex angles are 30 degrees, then each of their base angles must be equal to 75 degrees because base angles of an isosceles triangle are equal. you get:
angle AEB = 75 degrees.
angle EAB = 75 degrees.
angle DEC = 75 degrees.
angle EDC = 75 degrees.
the sum of the angles of each of those triangle is equal to 180 degrees as it should be.
2 * 75 + 30 = 150 + 30 = 180.
if you look at point E, the sum of the angles surrounding that point must be equal to 360 degrees.
you have already determined that:
angle AEB = 75 degrees.
angle BEC = 60 degrees.
angle DEC = 75 degrees.
the last angle to complete the circle is angle AED.
that angle has to be equal to 150 degrees because 75 + 60 + 75 + 150 = 360.
since that's the angle you wanted to find the measure of, you're done.
the following diagram shows you what i was talking about.
the single hash mark on all the sides tell you that they are all congruent to each other.
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