ABCD is square and CDE is an equilateral triangle outside the square. What is the value (in degrees) of `angleBEC` ?

Options

A. 15
B. 30
C. 25
D. 10

Answer

To solve for the value of angle BEC in the given configuration of square ABCD and equilateral triangle CDE, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Angles in the Equilateral Triangle**:
- Since CDE is an equilateral triangle, all its angles are equal to 60 degrees. Therefore, angle DCE = angle CED = angle CDE = 60 degrees.

**Hint**: Remember that in an equilateral triangle, all sides and angles are equal.

2. **Identify the Angles in the Square**:
- In square ABCD, all angles are right angles (90 degrees). Therefore, angle ABC = 90 degrees.

**Hint**: A square has four right angles, each measuring 90 degrees.

3. **Determine Angle BEC**:
- In triangle BEC, we know angle BEC is the angle we want to find, angle CBE is the angle formed by line BC and line BE, and angle CEB is the angle formed by line CE and line BE.
- From the previous steps, we know:
- Angle CBE = angle ABC + angle DCE = 90 degrees + 60 degrees = 150 degrees.

**Hint**: The angle CBE is the sum of the right angle from the square and the angle from the equilateral triangle.

4. **Use the Triangle Angle Sum Property**:
- The sum of the angles in triangle BEC is 180 degrees. Therefore:
\[ \text{Angle BEC} + \text{Angle CBE} + \text{Angle CEB} = 180 \text{ degrees} \]
- We already know angle CBE = 150 degrees and angle CEB = 60 degrees (since it is also part of the equilateral triangle).
- Plugging in the values:
\[ \text{Angle BEC} + 150 + 60 = 180 \]
\[ \text{Angle BEC} + 210 = 180 \]
\[ \text{Angle BEC} = 180 - 210 = -30 \text{ degrees} \]
- Since angles cannot be negative, we must have made an error in our assumptions about angle CEB. Let's correct that.

5. **Correct Calculation of Angle CEB**:
- Since angle CEB is actually the angle opposite to angle CBE, we can use the fact that angle CEB = 30 degrees (as it is the remaining angle in the triangle).
- Thus, we have:
\[ \text{Angle BEC} + 150 + 30 = 180 \]
\[ \text{Angle BEC} + 180 = 180 \]
\[ \text{Angle BEC} = 0 \text{ degrees} \]
- This indicates that angles were miscalculated. Let's recalculate angle BEC directly.

6. **Final Calculation**:
- Since angle BEC is actually the external angle at point E, we can use:
\[ \text{Angle BEC} = 180 - \text{Angle CBE} = 180 - 150 = 30 \text{ degrees} \]

### Conclusion:
The value of angle BEC is **30 degrees**.