https://file.aiquickdraw.com/mathbot/file/1724936204241y1qscyhy.jpg

This is an Octagon. Work out an expression for θ, giving your answer in terms of $$ x. $$

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.

Learn how to derive an expression for θ in an octagon, given various angle measurements including x. This problem involves calculating polygon angles and applying the sum of interior angles theorem. Suitable for students in Grades 10-12.

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