To find the angle $x$ in quadrilateral $ABCD$, we note the following:

• We know angle $C$ is $29^\circ$.
• Triangle $BCD$ can have its angles determined using the triangle sum theorem, which states that the sum of the interior angles of a triangle is $180^\circ$.

Thus, in triangle $BCD$: $\angle B + \angle D + \angle C = 180^\circ$ Given that $\angle C = 29^\circ$, and assuming $\angle B = x$ and $\angle D = 90^\circ$ (as indicated by the square in the corner at point $D$), we can set up the equation: $x + 90^\circ + 29^\circ = 180^\circ$

Let's solve this equation to find $x$.The value of angle $x$ in quadrilateral $ABCD$ is $61^\circ$.

Would you like more details on how this was calculated or have any other questions? Here are some related topics you might consider:

1. How can the triangle sum theorem be used to solve other problems involving polygons?
2. What are the properties of different types of quadrilaterals?
3. How would the method change if $D$ was not a right angle?
4. How to calculate the area of quadrilateral $ABCD$ using the given side lengths and angles?
5. What are the principles of trigonometry that could help solve for other unknowns in such shapes?

Tip: Always verify that a triangle's angles add up to $180^\circ$ and use this to check your work when calculating unknown angles.