The question involves an isosceles triangle and a quadrilateral. We are tasked with finding the unknown angle $x$ in the quadrilateral.

Given Information:

1. The triangle is isosceles, with angles of $30^\circ$ and $60^\circ$.
2. There is a quadrilateral attached to the triangle, and the angle $x$ in the quadrilateral is what we need to find.

Step-by-step Solution:

1. Isosceles Triangle Properties:

   • In the triangle, the sum of angles in any triangle is $180^\circ$.
   • You are given that one angle is $30^\circ$ and another is $60^\circ$.
   • The third angle of the triangle would be: $180^\circ - (30^\circ + 60^\circ) = 90^\circ$ Therefore, this is a right-angled triangle.

2. Quadrilateral Angle Sum:

   • The sum of angles in any quadrilateral is $360^\circ$.
   • Three of the angles are already given:
     • The bottom-left angle in the quadrilateral, adjacent to the triangle, is $90^\circ$ (from the triangle).
     • The two other angles of the quadrilateral are $30^\circ$ and $60^\circ$.

3. Finding $x$:

   • Let's sum up the known angles in the quadrilateral: $90^\circ + 30^\circ + 60^\circ = 180^\circ$
   • Since the total sum of the angles in a quadrilateral is $360^\circ$, the unknown angle $x$ is: $x = 360^\circ - 180^\circ = 180^\circ$

Thus, the value of angle $x$ is 180°.

Would you like further clarification or a detailed breakdown of any part? Here are some related questions you might want to consider:

1. What is the general formula for the sum of the interior angles of a polygon?
2. How does the sum of the angles in a triangle help in solving quadrilateral problems?
3. Why do right angles play a significant role in geometric proofs?
4. How can you prove that the given triangle is isosceles?
5. What other properties of quadrilaterals could be used to find unknown angles?

Tip: Remember, the sum of angles in a polygon with $n$ sides is $(n-2) \times 180^\circ$. This is useful for solving more complex geometric problems.