To find the value of $x$ in the given figure, let’s analyze the information provided.

1. The quadrilateral $ABCD$ has sides labeled, and a diagonal $AC$ divides it into two triangles.
2. We are given:
   • $\angle CAB = 35^\circ$
   • $\angle ABC = 92^\circ$
   • The lengths $AB = 3.4$, $CD = 3.4$, $BC = 4.7$, and $AD = 4.7$.
3. We need to find $x^\circ$, which is the measure of $\angle CDB$.

Step-by-Step Solution

Since $ABCD$ is a quadrilateral, and specifically, appears to be a parallelogram due to opposite sides being equal (i.e., $AB = CD$ and $AD = BC$), we can apply the following properties:

1. Opposite Angles in a Parallelogram: Opposite angles are equal, and consecutive angles are supplementary (sum up to $180^\circ$).

2. Using the Triangle $\triangle ACB$:

   • Since $\angle CAB = 35^\circ$ and $\angle ABC = 92^\circ$, we can find $\angle BCA$: $\angle BCA = 180^\circ - \angle CAB - \angle ABC = 180^\circ - 35^\circ - 92^\circ = 53^\circ$

3. Finding $x$:

   • Since $\angle BCA = 53^\circ$, and opposite angles in a parallelogram are equal, $x = 53^\circ$.

Answer

$x = 53^\circ$

Would you like more detailed reasoning or further explanation on any part of this solution?

1. How would the answer change if the shape wasn’t a parallelogram?
2. Can we use any other methods to verify that this is indeed a parallelogram?
3. What properties of parallelograms are essential for solving this problem?
4. Are there any specific theorems applicable to diagonals in parallelograms?
5. How can we solve similar problems involving unknown angles in other quadrilaterals?

Tip: Always check if a quadrilateral has parallel opposite sides or equal opposite angles to identify it as a parallelogram.