To solve the problem of expressing $\theta$ in terms of $x$ in the given octagon, we can follow these steps:

1. Sum of Interior Angles in an Octagon

The sum of the interior angles of an octagon (8-sided polygon) can be calculated using: $\text{Sum of interior angles} = (n-2) \times 180^\circ$ For an octagon, $n = 8$: $\text{Sum of interior angles} = (8-2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ$

2. Identify the Given Angles

From the image, the following angles are labeled:

• $220^\circ$
• $120^\circ$
• $\theta$
• Two angles are $x$
• One angle is labeled $(x - 40^\circ)$

3. Set Up the Equation

Since we are given six angles and need to express $\theta$ in terms of $x$, sum all these angles and set it equal to $1080^\circ$:

$220^\circ + 120^\circ + \theta + x + x + (x - 40^\circ) = 1080^\circ$

Simplify the equation: $2x + (x - 40^\circ) + 340^\circ + \theta = 1080^\circ$ $3x + 300^\circ + \theta = 1080^\circ$

4. Solve for $\theta$

Isolate $\theta$: $\theta = 1080^\circ - 3x - 300^\circ$ $\theta = 780^\circ - 3x$

Conclusion

The expression for $\theta$ in terms of $x$ is: $\theta = 780^\circ - 3x$

Would you like further details or have any questions? Here are some additional related questions:

1. How would the equation change if one of the angles is unknown?
2. Can you express $x$ in terms of $\theta$?
3. What is the measure of $\theta$ if $x = 200^\circ$?
4. How do the properties of an octagon help in solving for unknown angles?
5. What would happen if this octagon was regular (all sides and angles are equal)?

Tip: When working with polygons, understanding the relationship between the sum of interior angles and the number of sides can make solving for unknown angles much easier.