https://file.aiquickdraw.com/mathbot/file/1725887038006ic32epkf.jpg

To solve the problem shown in the image, where you need to calculate the measure of each lettered angle, let’s analyze the given figures:

### Figure 3 (Hexagon with angles labeled and letters `s`, `t`, `o`):
1. **Understanding the angles**:
   - The sum of the interior angles of a hexagon is calculated using the formula:
     \[
     \text{Sum of interior angles} = (n - 2) \times 180^\circ
     \]
     where \( n \) is the number of sides. For a hexagon, \( n = 6 \), so:
     \[
     \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 720^\circ
     \]
   - The given angles are 112°, 115°, 121°, 117°, and 130°. Let the unknown angles \( s \), \( t \), and \( o \) be represented by \( s^\circ \), \( t^\circ \), and \( o^\circ \) respectively.

2. **Sum of the known angles**:
   \[
   112^\circ + 115^\circ + 121^\circ + 117^\circ + 130^\circ = 595^\circ
   \]

3. **Finding the unknown angle `t`**:
   Since \( t \) is the remaining interior angle in the hexagon:
   \[
   t^\circ = 720^\circ - 595^\circ = 125^\circ
   \]
   Therefore, \( t = 125^\circ \).

### Figure 4 (Regular Hexagon with letter `m`):
1. **Understanding the regular hexagon**:
   - A regular hexagon has equal sides and equal interior angles. Each interior angle of a regular hexagon is calculated as:
     \[
     \text{Interior angle} = \frac{720^\circ}{6} = 120^\circ
     \]

2. **Finding the exterior angle `m`**:
   - The exterior angle \( m \) can be calculated by subtracting the interior angle from 180°:
     \[
     m^\circ = 180^\circ - 120^\circ = 60^\circ
     \]

### Summary of Angles:
- \( s = 117^\circ \)
- \( t = 125^\circ \)
- \( o = 130^\circ \)
- \( m = 60^\circ \)

Would you like to explore the steps in more detail or have any other questions?

### Follow-up Questions:
1. How would you find the sum of interior angles for a polygon with 8 sides?
2. Can you calculate the exterior angles of a regular decagon?
3. How would the interior angle change if the hexagon in Figure 3 were irregular?
4. What would be the measure of angle \( t \) if angle \( o \) were increased by 10°?
5. How can you use symmetry to simplify calculations in regular polygons?

### Tip:
Remember that for any regular polygon, the exterior angles always sum up to 360°, regardless of the number of sides!

Calculate the measure of each lettered angle in the hexagon and regular hexagon.

Solve for the lettered angles in a hexagon and a regular hexagon, using geometry concepts such as the sum of interior angles and exterior angles. Key angles include t = 125°, o = 130°, and m = 60°. Suitable for students in Grades 8-10.

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