In this figure, two parallelograms $ABDH$ and $CEFG$ are given, and we are asked to find the value of $x$.

Step-by-step Solution:

1. The point $O$ is the intersection of the diagonals of both parallelograms.

2. In a parallelogram, opposite angles are equal, and the diagonals bisect each other. This means that:

   • The angle $\angle BCD$ and $\angle DCE$ should be equal, as diagonals bisect the angles at the intersection.
   • Both of these angles are labeled as $x$.

3. From the geometry of the diagram:

   • The angle at point $C$ is given as $130^\circ$.
   • The external angle at $F$, i.e., $\angle GFE = 30^\circ$.

Using angle properties:

At point $O$, we have: $\angle BCD + \angle DCE + \angle BCD = 360^\circ.$ Since both angles $BCD$ and $DCE$ are equal to $x$, we can write: $2x + 130^\circ = 180^\circ.$

Solving for $x$:

$2x = 180^\circ - 130^\circ = 50^\circ,$ $x = \frac{50^\circ}{2} = 25^\circ.$

Thus, the value of $x$ is $25^\circ$.

Would you like more details or have any further questions?

Related Questions:

1. How do we use properties of parallelograms to find missing angles?
2. How are diagonals of parallelograms useful in geometric proofs?
3. What are the rules for external angles in polygons like parallelograms?
4. How can bisected angles at diagonals help solve for unknown values?
5. Can you generalize the solution method for other parallelograms with different angle values?

Tip:

Always remember that in a parallelogram, opposite angles are equal and diagonals bisect each other. This is useful when solving for unknown angles at diagonal intersections.