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The problem involves quadrilateral ABCD, where:

- \( AB = AD \)
- \( BD = BC \)
- \( DCE \parallel AB \)
- \( \hat{A} = 34^\circ \)

We are tasked with calculating \( \hat{B_2} \), the second angle at point B.

### Step-by-step Solution:

1. Since \( AB = AD \) and \( BD = BC \), triangle ABD is isosceles, and so is triangle BCD. This means:
   \[
   \hat{A_1} = \hat{D_1} \quad \text{(since triangle ABD is isosceles)}
   \]
   \[
   \hat{B_2} = \hat{C_2} \quad \text{(since triangle BCD is isosceles)}
   \]
   
2. The angle \( \hat{A} = 34^\circ \) is split as follows:
   \[
   \hat{A_1} = \hat{A_2} = 34^\circ / 2 = 17^\circ
   \]
   
3. Since \( DCE \parallel AB \), by the alternate interior angles theorem, \( \hat{D_2} = \hat{A_2} = 17^\circ \).

4. In triangle ABD, the sum of angles must be \( 180^\circ \):
   \[
   \hat{A_1} + \hat{D_1} + \hat{B_1} = 180^\circ
   \]
   Since \( \hat{A_1} = \hat{D_1} = 17^\circ \), we can solve for \( \hat{B_1} \):
   \[
   17^\circ + 17^\circ + \hat{B_1} = 180^\circ
   \]
   \[
   \hat{B_1} = 180^\circ - 34^\circ = 146^\circ
   \]
   
5. Now, in triangle BCD, we know \( \hat{D_2} = 17^\circ \). Since triangle BCD is isosceles and \( \hat{B_2} = \hat{C_2} \), the sum of angles in triangle BCD gives:
   \[
   \hat{B_2} + \hat{C_2} + \hat{D_2} = 180^\circ
   \]
   Since \( \hat{B_2} = \hat{C_2} \), we can express this as:
   \[
   2\hat{B_2} + 17^\circ = 180^\circ
   \]
   \[
   2\hat{B_2} = 163^\circ
   \]
   \[
   \hat{B_2} = \frac{163^\circ}{2} = 81.5^\circ
   \]

Thus, the size of \( \hat{B_2} \) is \( 81.5^\circ \).

Would you like more details or have any questions about this solution?

Here are some related questions to explore:

1. How do parallel lines affect angle relationships in quadrilaterals?
2. What are the properties of isosceles triangles, and how are they useful in solving geometry problems?
3. Can this problem be solved using the Law of Sines or Cosines? If so, how?
4. How does the concept of symmetry help in solving geometric problems?
5. What other methods can be used to solve angle problems in cyclic quadrilaterals?

**Tip:** Always remember that the sum of the interior angles of any triangle is \( 180^\circ \), which is crucial for solving many geometric problems!

In quadrilateral ABCD, AB = AD, BD = BC, DCE is parallel to AB, and angle A is 34°. Calculate the size of angle B2.

Learn how to calculate angle B2 in a quadrilateral where AB = AD, BD = BC, and DCE is parallel to AB. This problem involves concepts like isosceles triangles, parallel lines, and angle sum properties. Step-by-step solution provided for students in Grades 9-10.

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