The problem involves a rectangle $ABCD$ with the diagonals $AC$ and $BD$ intersecting at point $O$. You're asked to find the value of $x$ given that $\angle EQR = 40^\circ$.

However, the image resolution is low, and the exact details around point $E$, $Q$, and $R$ are unclear. But, based on common rectangle properties and the diagram structure, we can consider the following:

Key Properties of Rectangles:

1. In a rectangle, the diagonals are equal in length and bisect each other.
2. Each angle in a rectangle is $90^\circ$.
3. The diagonals of a rectangle form two congruent triangles.

Given:

• $\angle EQR = 40^\circ$.
• Since $ABCD$ is a rectangle, the diagonal $AC$ and $BD$ intersect at $O$, forming congruent triangles $\triangle AOB$, $\triangle BOC$, $\triangle COD$, and $\triangle DOA$.

To solve for $x$ (which likely represents an angle or segment in the diagram):

1. Use the fact that opposite angles in the triangles formed by the diagonals in the rectangle are congruent.
2. If $x$ represents an angle, understand that in any right-angled triangle with one angle $40^\circ$, the remaining angle can be found using the fact that the sum of angles in a triangle is $180^\circ$.

Solution Approach:

• If $\angle EQR$ is part of a triangle where the other angle is $x$, then: $x = 180^\circ - 40^\circ - 90^\circ$ $x$ would then be $50^\circ$.

But without further clarification from the image, I suggest confirming the exact relationship or location of $EQR$ within the rectangle, as it may impact the exact value.

If the problem is set up as expected, the value for $x$ could be $50^\circ$, assuming it forms a triangle with the given angles.

Would you like further details or clarification? Here are 5 related questions you might consider:

1. How do you find the angles formed by the diagonals of a rectangle?
2. What properties of rectangles are useful in solving angle problems?
3. How does the sum of angles in a triangle help in solving problems like this?
4. What are some real-life applications of understanding rectangle properties?
5. How would the problem change if the shape were a different quadrilateral, like a parallelogram?

Tip: When working with diagrams, always verify all given angles and segment lengths carefully before starting your calculations.