Find the value of θ

• We have two isosceles triangles: $\triangle ABD$ and $\triangle BCD$.
• In $\triangle ABD$, the double hash marks indicate that $AD = BD$.
• In $\triangle BCD$, the single hash marks indicate that $BC = CD$.
• The angle $\angle ADB = 52^\circ$.
• The angle we need to find is $\theta = \angle BCD$.

Since $\triangle ABD$ is an isosceles triangle with $AD = BD$, the angles opposite these sides are equal: $\angle BAD = \angle ABD$.
The sum of angles in a triangle is $180^\circ$.

So, $\angle BAD = 64^\circ$ and $\angle ABD = 64^\circ$.

Since $\triangle BCD$ is an isosceles triangle with $BC = CD$, the angles opposite these sides are equal: $\angle CBD = \angle CDB$.
Let $\angle CBD = \angle CDB = x$.
The sum of angles in $\triangle BCD$ is $180^\circ$.

The angles of $\triangle ACD$ are:

• $\angle CAD = \angle BAD = 64^\circ$ (from Step 2).
• $\angle ACD = \theta$ (given).
• $\angle ADC = \angle ADB + \angle CDB = 52^\circ + x$ (where $x = \angle CDB$ from Step 3). The sum of angles in $\triangle ACD$ is $180^\circ$.

We have two equations:

1. $\theta + 2x = 180^\circ$ (from $*$)
2. $\theta + x = 64^\circ$ (from $**$)

Subtract equation (2) from equation (1):

Now substitute the value of $x$ into equation (2):

The result $\theta = -52^\circ$ is not possible for an interior angle of a triangle. This indicates that the geometric configuration as depicted and interpreted with standard Euclidean geometry rules is impossible. There might be an error in the problem statement (e.g., incorrect angle value, incorrect side markings, or a non-Euclidean context). Assuming the problem expects a positive angle, this specific setup leads to a contradiction.

Given the constraints of a typical geometry problem, it's highly probable that there's a mislabeling in the diagram. If we assume the angle $52^\circ$ was intended to be $\angle BDC$ instead of $\angle ADB$, then:
If $\angle BDC = 52^\circ$:
In $\triangle BCD$, $BC=CD \implies \angle CBD = \angle CDB = 52^\circ$.
Then $\theta = \angle BCD = 180^\circ - 52^\circ - 52^\circ = 180^\circ - 104^\circ = 76^\circ$.
However, the angle is clearly marked as $\angle ADB$.

Based on the diagram as presented, the calculation leads to an impossible angle. If this is a multiple-choice question, none of the options would be correct. If it's a free-response question, the answer would be that the figure is impossible as labeled.

Since I must provide a numerical answer, and the direct interpretation leads to an impossible result, I cannot provide a valid positive angle for $\theta$ under standard geometric rules. There might be an error in the problem itself.