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Let's solve each of these questions step by step:

### Question 6
We have three triangles, and we need to find the unknowns.

#### Part (a)
Given:
- One side: \( 8 \text{ cm} \)
- Angles: \( 52^\circ \) and \( 43^\circ \)

This is a non-right triangle, so we'll use the **Sine Rule**:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
We can find the third angle: \( 180^\circ - 52^\circ - 43^\circ = 85^\circ \).

Let’s assign:
- \( a = 8 \text{ cm}, A = 43^\circ \)
- \( b = x \text{ cm}, B = 52^\circ \)

\[
\frac{8}{\sin 43^\circ} = \frac{x}{\sin 52^\circ}
\]

Solving for \( x \):
\[
x = \frac{8 \cdot \sin 52^\circ}{\sin 43^\circ}
\]

#### Part (b)
Given:
- Sides: \( 10 \text{ m}, 11 \text{ m}, 12 \text{ m} \)

We use the **Cosine Rule** to find angle \( \theta \):
\[
c^2 = a^2 + b^2 - 2ab\cos C
\]
Let:
- \( a = 10 \text{ m}, b = 11 \text{ m}, c = 12 \text{ m} \)

\[
12^2 = 10^2 + 11^2 - 2 \cdot 10 \cdot 11 \cdot \cos \theta
\]

Solve for \( \theta \).

#### Part (c)
Given:
- Side lengths: \( 4 \text{ cm}, 9 \text{ cm} \)
- Angle: \( 37^\circ \)

Using the **Cosine Rule** again:
\[
c^2 = a^2 + b^2 - 2ab\cos C
\]
Let:
- \( a = 4 \text{ cm}, b = 9 \text{ cm}, C = 37^\circ \)

Solve for the unknown side \( x \).

### Question 7
To determine whether to use the Sine Rule or the Cosine Rule:
- If you know two sides and the angle between them (SAS), use the **Cosine Rule**.
- If you know a side and its opposite angle along with another side or angle, use the **Sine Rule**.

### Question 8
Find the area of quadrilateral \( ABCD \). The quadrilateral can be divided into two triangles using diagonal \( BD \). We use the formula:
\[
\text{Area} = \frac{1}{2}ab\sin C
\]
Calculate the areas of \( \triangle ABD \) and \( \triangle BCD \) separately.

### Question 9
For the triangle:
- Sides: \( 9 \text{ cm}, 12 \text{ cm}, 17 \text{ cm} \)

Apply **Heron's Formula**:
\[
s = \frac{a + b + c}{2}
\]
\[
\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
\]

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Would you like a detailed solution for a specific question or more explanations? 

### Related Questions:
1. What is the Sine Rule and when is it applied?
2. How does the Cosine Rule differ when finding angles versus side lengths?
3. How can Heron's Formula be used for any triangle?
4. What other methods can be used to find the area of a quadrilateral?
5. Why is the sum of angles in a triangle always \( 180^\circ \)?

### Tip:
Always verify if a triangle is right-angled before applying the Sine or Cosine Rule; sometimes the Pythagorean theorem might be simpler.

Find the unknowns in triangles using the Sine and Cosine rules, determine the appropriate rule for given data, and calculate areas of quadrilaterals and triangles.

Learn how to solve unknowns in triangles using the Sine and Cosine rules, and find the areas of quadrilaterals and triangles. This covers advanced trigonometry and geometry concepts suitable for high school students in Grades 10-12.

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