Let's analyze the new image to find the value of $x$.

Description and Analysis:

The image depicts a quadrilateral with the following angles labeled:

• One angle is $x^\circ$ (the value we need to find).
• Another angle is $50^\circ$.
• There is a right angle ($90^\circ$).
• The fourth angle is $120^\circ$.

Solution:

The sum of the interior angles of any quadrilateral is always $360^\circ$. We can set up the equation for the sum of the angles in this quadrilateral.

1. Sum of the angles: $x^\circ + 50^\circ + 90^\circ + 120^\circ = 360^\circ$
2. Combine the known angles: $x^\circ + 260^\circ = 360^\circ$
3. Solving for $x$: $x^\circ = 360^\circ - 260^\circ = 100^\circ$

Thus, the value of $x$ is $100^\circ$.

Would you like more details or have any additional questions?

Follow-up Questions:

1. How would the calculation change if this were a pentagon instead of a quadrilateral?
2. Why is the sum of the interior angles of a quadrilateral always $360^\circ$?
3. Can the sum of the angles in a quadrilateral ever be different from $360^\circ$?
4. How would you verify your answer using a protractor?
5. What are the properties of special types of quadrilaterals like rectangles and squares regarding their angles?

Tip:

When solving for unknown angles in polygons, always start by recalling the sum of the interior angles for that shape. For a quadrilateral, it's $360^\circ$, which is fundamental to solving for unknown angles.