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Use front left corner as the Coordinate origin

To calculate the required angles in this problem using the coordinate system where the front left corner is the origin, we will first extract the necessary geometric details and proceed step-by-step for each part (a, b, and c).

### Given:
- The height of the house wall is 3.0 m.
- The height of the roof peak is 4.4 m.
- The house dimensions are: 
  - 8.8 m long
  - 7.6 m wide.

The dimensions and angles involve right triangle trigonometry and 3D spatial geometry. We can begin with each question:

### Part a: Angle between the two roof surfaces
The roof forms a ridge in the middle of the house, and each side of the roof slopes down from this central ridge to the edges of the house. The question asks for the angle between the two sloped roof surfaces.

To calculate the angle between the two roof planes:
1. The vertical difference between the roof's ridge and the wall's height is \(4.4\,\text{m} - 3.0\,\text{m} = 1.4\,\text{m}\).
2. The half-width of the house is \(7.6\,\text{m} / 2 = 3.8\,\text{m}\).

We can now calculate the slope of the roof using the tangent of the roof's angle on one side. 
\[
\tan(\theta_1) = \frac{1.4\,\text{m}}{3.8\,\text{m}}
\]
This gives the angle of one roof plane with respect to the horizontal. The angle between the two roof planes will be twice this value (since they are symmetric):

\[
\theta_{a} = 2\theta_1
\]

### Part b: Angle between one roof surface and the house wall
This angle is simply the slope of one side of the roof. It is the same as the value \( \theta_1 \) calculated in part (a).

### Part c: Angle between one edge of the chimney and the roof surface
The diagram shows that the chimney is perpendicular to the roof, meaning the angle between the chimney edge and the roof surface is \(90^\circ\).

I will now calculate these values.Here are the calculated angles:

- **a) The angle between the two roof surfaces**: \( 40.45^\circ \)
- **b) The angle between one roof surface and the house wall**: \( 20.22^\circ \)
- **c) The angle between one edge of the chimney and the roof surface**: \( 90^\circ \) (as the chimney is perpendicular to the roof)

Would you like more details on any of these calculations, or do you have any additional questions?

Here are 5 related questions for further exploration:
1. How would the angles change if the house were wider or narrower?
2. What trigonometric relationships are used in determining roof slopes?
3. How can you calculate the surface area of the roof using these angles?
4. What effect would raising the roof ridge have on the slope of the roof?
5. How does the length of the house impact the overall roof structure?

**Tip:** Always remember that the tangent function \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) is useful for calculating angles in right triangles!

Calculate the angles related to the roof structure: a) between the two roof surfaces, b) between a roof surface and the house wall, c) between the chimney and the roof surface.

Learn how to calculate angles in a house structure involving the roof surfaces, the house wall, and a chimney. This problem involves trigonometry, calculating slopes, and using right triangle geometry. Detailed solutions are provided, suitable for high school students in grades 9-12.

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