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!