The problem asks us to find the value of $x$ and then determine the missing angle measures of the polygon shown. Here's how we can solve it:

Step 1: Set Up the Equation

The polygon is a pentagon, and the sum of its interior angles is given as $540^\circ$. The interior angles are labeled as $x$, $x$, $x-30^\circ$, $x-30^\circ$, and $x-30^\circ$.

Step 2: Write the Equation for the Sum of the Angles

We add all the angles together and set them equal to $540^\circ$:

$x + x + (x - 30) + (x - 30) + (x - 30) = 540$

Simplify the equation:

$5x - 90 = 540$

Step 3: Solve for $x$

Add 90 to both sides:

$5x = 630$

Divide by 5:

$x = 126$

Step 4: Find the Missing Angle Measures

Substitute $x = 126$ into the expressions for each angle:

• For the angles labeled $x$: $x = 126^\circ$

• For the angles labeled $x - 30$: $x - 30 = 126 - 30 = 96^\circ$

Final Answers:

• $x = 126$
• Angles labeled $x - 30$ are $96^\circ$.

Would you like further details on any of the steps? Here are some related questions to expand on:

1. How do you determine the sum of the interior angles of any polygon?
2. What are some strategies for solving equations with multiple variables?
3. How do you identify and label angles in polygons with more than five sides?
4. How does changing one angle affect the overall sum in polygons?
5. What are some common mistakes to avoid when solving for unknown angles in polygons?

Tip: Always double-check the sum of the angles to ensure it matches the given total for the polygon's interior angles!