In the pentagon A, B, C, D, E AB=AE=12, CD=DE=3 and BC=6. Which of the following can be the value(s) of the `angleBAE` ?

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. All of these

Answer

To find the possible values of the angle \( \angle BAE \) in the pentagon \( A, B, C, D, E \) with the given side lengths, we can follow these steps:

### Step 1: Understand the given information
We have a pentagon with the following sides:
- \( AB = AE = 12 \)
- \( CD = DE = 3 \)
- \( BC = 6 \)

### Step 2: Identify the triangle
We need to focus on triangle \( ABE \) since we are looking for \( \angle BAE \). In triangle \( ABE \):
- \( AB = 12 \)
- \( AE = 12 \)
- The triangle is isosceles because \( AB = AE \).

### Step 3: Use the triangle angle sum property
The sum of the angles in triangle \( ABE \) is \( 180^\circ \):
\[ \angle A + \angle B + \angle E = 180^\circ \]

Since \( AB = AE \), we have \( \angle B = \angle E \). Let’s denote \( \angle B = \angle E = x \).

### Step 4: Set up the equation
Substituting into the angle sum equation:
\[ \angle A + 2x = 180^\circ \]
This implies:
\[ \angle A = 180^\circ - 2x \]

### Step 5: Determine the range for \( x \)
Since \( \angle A \) must be positive, we have:
\[ 180^\circ - 2x > 0 \implies 2x < 180^\circ \implies x < 90^\circ \]

### Step 6: Calculate possible values of \( x \)
Now, we can find possible values for \( x \):
- If \( x = 45^\circ \):
\[ \angle A = 180^\circ - 2(45^\circ) = 90^\circ \]

- If \( x = 60^\circ \):
\[ \angle A = 180^\circ - 2(60^\circ) = 60^\circ \]

- If \( x = 90^\circ \):
\[ \angle A = 180^\circ - 2(90^\circ) = 0^\circ \text{ (not possible)} \]

### Step 7: Conclusion
Thus, the possible values for \( \angle BAE \) (which is \( x \)) can be:
- \( 45^\circ \)
- \( 60^\circ \)
- Any value less than \( 90^\circ \)