In a 7-sided polygon, some of the interior angles are 124° each, and the remaining angles are x° each, where x is a multiple of 10. Find the value of x.

To solve this problem, we can follow these steps:

### Step 1: Calculate the sum of the interior angles of a 7-sided polygon.
The formula to calculate the sum of the interior angles of a polygon with \( n \) sides is:
\[
\text{Sum of interior angles} = 180^\circ \times (n - 2)
\]
For a 7-sided polygon:
\[
\text{Sum of interior angles} = 180^\circ \times (7 - 2) = 180^\circ \times 5 = 900^\circ
\]

### Step 2: Set up the equation for the angles.
Let the number of angles that measure 124° be \( k \). Therefore, the number of angles that measure \( x^\circ \) (a multiple of 10) will be \( 7 - k \).

The sum of the interior angles is then given by:
\[
k \times 124^\circ + (7 - k) \times x^\circ = 900^\circ
\]

### Step 3: Simplify the equation.
Expand and rearrange the equation:
\[
124k + (7 - k)x = 900
\]
\[
124k + 7x - kx = 900
\]
\[
k(124 - x) + 7x = 900
\]

### Step 4: Test for values of \( k \) and solve for \( x \).
Since \( x \) is a multiple of 10, we will substitute potential values of \( x \) and solve for \( k \) to ensure \( k \) is an integer.

Let's start testing for multiples of 10:

1. **If \( x = 120^\circ \)**:
    \[
    k(124 - 120) + 7(120) = 900
    \]
    \[
    4k + 840 = 900
    \]
    \[
    4k = 60
    \]
    \[
    k = 15 \, \text{(not possible since \( k \) cannot be greater than 7)}
    \]

2. **If \( x = 110^\circ \)**:
    \[
    k(124 - 110) + 7(110) = 900
    \]
    \[
    14k + 770 = 900
    \]
    \[
    14k = 130
    \]
    \[
    k = \frac{130}{14} \approx 9.29 \, \text{(not an integer)}
    \]

3. **If \( x = 100^\circ \)**:
    \[
    k(124 - 100) + 7(100) = 900
    \]
    \[
    24k + 700 = 900
    \]
    \[
    24k = 200
    \]
    \[
    k = \frac{200}{24} \approx 8.33 \, \text{(not an integer)}
    \]

4. **If \( x = 130^\circ \)**:
    \[
    k(124 - 130) + 7(130) = 900
    \]
    \[
    k(-6) + 910 = 900
    \]
    \[
    -6k = -10
    \]
    \[
    k = \frac{10}{6} \approx 1.67 \, \text{(not an integer)}
    \]

5. **If \( x = 140^\circ \)**:
    \[
    k(124 - 140) + 7(140) = 900
    \]
    \[
    -16k + 980 = 900
    \]
    \[
    -16k = -80
    \]
    \[
    k = 5 \, \text{(an integer and possible!)}
    \]

### Step 5: Verification
If \( k = 5 \) and \( x = 140^\circ \):
\[
5(124) + 2(140) = 620 + 280 = 900^\circ
\]
This satisfies the total sum of the angles.

Thus, the correct value of \( x \) is \( \boxed{140^\circ} \).

---

Would you like any additional clarification? Here are some follow-up questions:

1. How would the calculation change for a different polygon?
2. What if \( x \) was not a multiple of 10?
3. Can you generalize the method for any \( n \)-sided polygon?
4. How would this change if there were more or fewer angles of 124°?
5. How does the requirement for \( x \) being a multiple of 10 simplify the problem?

**Tip**: When solving for multiple unknowns with constraints, test values systematically to avoid missing possible solutions.

Learn how to calculate the interior angles of a 7-sided polygon where some angles are known and others are variables. This problem involves applying the formula for the sum of interior angles of a polygon and solving for unknown angles using algebraic equations. Suitable for students in Grades 9-10.

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