Let's solve each polygon separately to find the angle marked $x$.

Left Polygon

1. The sum of interior angles for a polygon with $n$ sides is calculated using the formula: $\text{Sum of interior angles} = (n - 2) \times 180^\circ$ This polygon has 7 sides, so: $\text{Sum of angles} = (7 - 2) \times 180^\circ = 5 \times 180^\circ = 900^\circ$

2. Add up the given angles: $71^\circ + 142^\circ + 125^\circ + 31^\circ + 143^\circ + 45^\circ + 105^\circ = 662^\circ$

3. Set up the equation to solve for $x$: $x + 662^\circ = 900^\circ$ $x = 900^\circ - 662^\circ = 238^\circ$

Thus, $x = 238^\circ$ for the left polygon.

Right Polygon

1. This polygon has 5 sides, so the sum of its interior angles is: $\text{Sum of angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ$

2. Add up the given angles: $20^\circ + 86^\circ + 34^\circ + 97^\circ = 237^\circ$

3. Set up the equation to solve for $x$: $x + 237^\circ = 540^\circ$ $x = 540^\circ - 237^\circ = 303^\circ$

Thus, $x = 303^\circ$ for the right polygon.

Final Answer:

• Left Polygon: $x = 238^\circ$
• Right Polygon: $x = 303^\circ$

Would you like further details or have any questions about this solution?

Here are some related questions:

1. How do you calculate the sum of interior angles for any polygon?
2. Why is the formula for the sum of interior angles $(n - 2) \times 180^\circ$?
3. How do you solve for an unknown angle when given multiple angles in a polygon?
4. What is the difference between interior and exterior angles in polygons?
5. Can you apply these methods to irregular polygons with different side lengths?

Tip: Always double-check the number of sides in a polygon to ensure you're using the correct formula for angle sums.