ABCD is a square and ABE is an equilateral triangle outside the square then

Options

A. `angleACE=(1)/(2) angleABE`
B. `angleACE= angleABE`
C. `angleACE=2 angleABE`
D. None of these

Answer

To solve the problem involving the square ABCD and the equilateral triangle ABE, we will follow these steps:

### Step 1: Understanding the Shapes
We have a square ABCD and an equilateral triangle ABE constructed outside the square.

### Step 2: Identify the Properties of the Shapes
1. In a square, all sides are equal and all angles are 90 degrees.
2. In an equilateral triangle, all sides are equal and all angles are 60 degrees.

### Step 3: Assigning Angles
- Let’s denote the angles of triangle ABE:
- Angle ABE = 60 degrees
- Angle BAE = 60 degrees
- Angle AEB = 60 degrees

### Step 4: Finding the Angles of the Square
- Since ABCD is a square:
- Angle ABC = 90 degrees
- Angle BCD = 90 degrees
- Angle CDA = 90 degrees
- Angle DAB = 90 degrees

### Step 5: Relationship Between the Angles
- Since ABE is an equilateral triangle, we have:
- AB = BE (sides of the equilateral triangle)
- Since ABCD is a square, we have:
- AB = BC (sides of the square)

### Step 6: Conclusion
From the above relationships, we can conclude:
- Since AB = BE and AB = BC, it follows that BC = BE.
- Therefore, triangle ABE is congruent to triangle ABC in terms of side lengths.

### Step 7: Finding Angle AEC
To find angle AEC, we can use the property of angles in a triangle:
- Since angle ABE = 60 degrees and angle ABC = 90 degrees, angle AEC can be calculated as follows:
- Angle AEC = Angle ABC - Angle ABE
- Angle AEC = 90 degrees - 60 degrees = 30 degrees.

### Final Answer
Thus, angle AEC = 30 degrees.

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