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Let's go through each question one by one to solve them.

### Question 1: Wood for the house gable

**a) Angle of the roof slope**  
The figure shows the dimensions of the roof (16.4 m base and 14.0 m height). We need to calculate the roof's angle \(\alpha\).

\[
\tan(\alpha) = \frac{\text{height}}{\text{base}} = \frac{14.0}{16.4}
\]

\[
\alpha = \tan^{-1} \left( \frac{14.0}{16.4} \right)
\]

Let's calculate that.

**b) How much wood is needed with 8% excess**  
First, we calculate the area of the gable using the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16.4 \times 14.0
\]

After calculating the area, we'll add an 8% allowance for wastage.

**c) Cost of wood including tax**  
Given the area in square meters and the cost per square meter (€36), we can calculate the total cost. We'll also add 19% VAT.

### Question 2: Isosceles triangle ABC with base \(AB\)

We have an isosceles triangle with:

- \(b = AB = 5.2 \, \text{cm}\) (base),
- \(h_c = 4.8 \, \text{cm}\) (height from \(C\) to \(AB\)).

**a) Find the missing sides and angles**  
First, we split the triangle in half using the height, giving two right-angled triangles. The legs are:

\[
\frac{b}{2} = \frac{5.2}{2} = 2.6 \, \text{cm}
\]

Using the Pythagorean theorem, we can find the equal sides \(AC = BC\):

\[
AC = BC = \sqrt{\left(\frac{b}{2}\right)^2 + h_c^2}
\]

Then, we'll calculate the angles using trigonometric ratios.

**b) Calculate the area**  
The area of the triangle is:

\[
\text{Area} = \frac{1}{2} \times b \times h_c
\]

### Question 3: Perimeter and area of the irregular quadrilateral

The figure shows a quadrilateral with side lengths:

- \(1.7 \, \text{cm}\), \(2.9 \, \text{cm}\), \(3.9 \, \text{cm}\), and \(1.7 \, \text{cm}\).

To find the area, we can divide the shape into simpler parts (triangles, trapezoids, or rectangles) and compute each part's area.

Let me calculate these values for you.Here are the calculations:

### Question 1: Wood for the house gable

**a) Angle of the roof slope**  
The angle \(\alpha\) of the roof slope is approximately:

\[
\alpha = 40.49^\circ
\]

**b) Area of wood needed with 8% excess**  
The area of the gable without wastage is:

\[
\text{Area} = 114.8 \, \text{m}^2
\]

With 8% wastage, the total wood needed is:

\[
123.98 \, \text{m}^2
\]

**c) Total cost of the wood including VAT**  
The total cost for the wood, including 19% VAT, is approximately:

\[
€5311.47
\]

---

### Question 2: Isosceles triangle \(ABC\)

**a) Missing sides**  
The two equal sides \(AC = BC\) are approximately:

\[
AC = BC = 5.46 \, \text{cm}
\]

**b) Area of the triangle**  
The area of the triangle is:

\[
\text{Area} = 12.48 \, \text{cm}^2
\]

---

Would you like further clarification or details on any part?  
Here are five related questions for further practice:

1. How would you calculate the roof's pitch percentage based on the angle?
2. If the height of the gable was increased by 1 meter, how would that affect the total wood needed?
3. How do you calculate the perimeter of the isosceles triangle \(ABC\)?
4. What trigonometric functions would you use to find the angles in the isosceles triangle \(ABC\)?
5. How would the cost of the wood change if the price per square meter increased by 10%?

**Tip**: For any triangle, knowing one height and the base lets you easily compute the area, which is a fundamental skill in geometry.

1. a) Calculate the angle of the roof slope for Mr. Targatz’s gable. b) Determine the required wood with an 8% allowance for wastage. c) Compute the cost of wood including 19% VAT. 2. ABC is an isosceles triangle. Calculate the missing sides, angles, and area. 3. Calculate the perimeter and area of an irregular quadrilateral by breaking it into simpler shapes.

Learn how to calculate the roof slope angle, determine the amount of wood needed with wastage, and compute the total cost including VAT. Also, solve for the missing sides and area of an isosceles triangle, and calculate the perimeter and area of an irregular quadrilateral.

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