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Let's solve each of these triangles one by one.

### Top-left triangle (first question):
Given:
- Angle 81°, Angle 31°
- Side opposite 31° = 3.5 cm
- Side opposite 81° = \(x\)

We will use the **sine rule**:

\[
\frac{x}{\sin 31^\circ} = \frac{3.5}{\sin 81^\circ}
\]

Now we solve for \(x\):

\[
x = \frac{3.5 \cdot \sin 31^\circ}{\sin 81^\circ}
\]

Let me calculate this for you.

### Top-right triangle (second question):
Given:
- Angle 78°, Angle 70°
- Side opposite 70° = 9.6 cm
- Side opposite 78° = \(x\)

Using the **sine rule** again:

\[
\frac{x}{\sin 70^\circ} = \frac{9.6}{\sin 78^\circ}
\]

Solving for \(x\):

\[
x = \frac{9.6 \cdot \sin 70^\circ}{\sin 78^\circ}
\]

I will calculate this as well.

### Bottom-left triangle (third question):
Given:
- Side 6.8 cm, Side 5.4 cm
- Angle between them = 42°
- Find angle \(x\)

We use the **cosine rule** to find the angle \(x\):

\[
\cos x = \frac{6.8^2 + 5.4^2 - 2 \cdot 6.8 \cdot 5.4 \cdot \cos 42^\circ}{2 \cdot 6.8 \cdot 5.4}
\]

I will calculate this next.

### Bottom-right triangle (fourth question):
Given:
- Side 7.5 cm, Side 6.4 cm
- Angle = 80°
- Find angle \(x\)

Using the **cosine rule**:

\[
\cos x = \frac{7.5^2 + 6.4^2 - 2 \cdot 7.5 \cdot 6.4 \cdot \cos 80^\circ}{2 \cdot 7.5 \cdot 6.4}
\]

Now, I'll calculate all values for you.Here are the results for each of the triangles:

### 1. Top-left triangle:
\[
x = 1.8 \, \text{cm}
\]

### 2. Top-right triangle:
\[
x = 9.2 \, \text{cm}
\]

### 3. Bottom-left triangle:
\[
x = 74^\circ \, (\text{rounded to the nearest degree})
\]

### 4. Bottom-right triangle:
\[
x = 33^\circ \, (\text{rounded to the nearest degree})
\]

Let me know if you need any further details or clarification. 

Here are some additional practice questions for you:
1. Solve a triangle using the sine rule when given two angles and one side.
2. How would you apply the cosine rule to find a missing side?
3. Can you calculate the area of a triangle when given two sides and the included angle?
4. What is the sum of angles in any triangle?
5. How does the sine rule differ from the cosine rule in applications?

**Tip**: Always check if a triangle problem can be simplified using the sum of angles before using trigonometric rules!

Find the side x and the angle x in each triangle using appropriate trigonometric rules.

Learn how to solve for sides and angles in acute-angled triangles using the sine and cosine rules. This tutorial includes step-by-step solutions for finding missing sides and angles, along with trigonometric formulas, and is perfect for students in Grades 9-12.

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