In the following polygons, the remaining angle have equal measure. Determine the measures. 70,____,____

Given that the remaining angles of the polygon have equal measures, let's first determine what type of polygon we're dealing with.

### Step 1: Identify the type of polygon
Let's assume the polygon has \( n \) sides. The sum of the interior angles of an \( n \)-sided polygon can be calculated using the formula:
\[
\text{Sum of interior angles} = 180^\circ \times (n - 2)
\]

### Step 2: Set up the equation
Given that one angle is \( 70^\circ \) and the other \( n-1 \) angles are equal, let the measure of each remaining angle be \( x \). The equation representing the sum of all angles in the polygon would be:
\[
70^\circ + (n-1)x = 180^\circ \times (n - 2)
\]

### Step 3: Solve for \( n \) and \( x \)
To find \( n \), let's test different values of \( n \) to see which yields a reasonable value for \( x \).

1. **For \( n = 4 \) (quadrilateral):**
   \[
   70^\circ + 3x = 180^\circ \times (4 - 2) = 360^\circ
   \]
   \[
   3x = 290^\circ \quad \Rightarrow \quad x = \frac{290^\circ}{3} \approx 96.67^\circ
   \]
   The angles are not equal.

2. **For \( n = 5 \) (pentagon):**
   \[
   70^\circ + 4x = 180^\circ \times (5 - 2) = 540^\circ
   \]
   \[
   4x = 470^\circ \quad \Rightarrow \quad x = \frac{470^\circ}{4} = 117.5^\circ
   \]
   The angles are still not equal.

3. **For \( n = 6 \) (hexagon):**
   \[
   70^\circ + 5x = 180^\circ \times (6 - 2) = 720^\circ
   \]
   \[
   5x = 650^\circ \quad \Rightarrow \quad x = \frac{650^\circ}{5} = 130^\circ
   \]
   The angles are now equal.

Thus, when \( n = 6 \) (a hexagon), the remaining angles have an equal measure of \( 130^\circ \).

### Final Answer:
The measures of the angles are:
\[
70^\circ, 130^\circ, 130^\circ, 130^\circ, 130^\circ, 130^\circ
\]

Would you like more details or have any questions?

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Here are some related questions to further explore:

1. How do you find the sum of the interior angles of any polygon?
2. What is the formula for the measure of each interior angle in a regular polygon?
3. How does the number of sides affect the measure of the interior angles of a polygon?
4. Can you explain why a hexagon's angles were found to be \( 130^\circ \)?
5. How would the calculation change if the initial angle were different?
6. How can you determine the type of polygon from its angle measures?
7. What are the exterior angles of a polygon, and how are they related to interior angles?
8. How can this method be applied to irregular polygons?

**Tip:** Always check the sum of interior angles and verify that the solution fits the criteria for the specific type of polygon.

Explore how to determine the measures of angles in polygons where remaining angles have equal measures. Learn the formula for the sum of interior angles of a polygon and see detailed step-by-step solutions. Suitable for Grades 9-12.

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