To solve these problems, we'll apply the Sine Rule, which is useful when we know two angles and one side, or two sides and one non-included angle.

Part 1: Calculating two possible values for angle $ACB$

We are given:

• $AB = 8.1 \, \text{cm}$
• $BC = 4.9 \, \text{cm}$
• $\angle BAC = 31^\circ$

Using the Sine Rule:

$\frac{AB}{\sin(\angle ACB)} = \frac{BC}{\sin(\angle BAC)}$

Let's solve for $\angle ACB$.

Step 1: Rearrange the Sine Rule to solve for $\sin(\angle ACB)$:

$\sin(\angle ACB) = \frac{AB \cdot \sin(\angle BAC)}{BC}$

Substitute the given values:

$\sin(\angle ACB) = \frac{8.1 \cdot \sin(31^\circ)}{4.9}$

Now, I'll compute this to get the value of $\sin(\angle ACB)$ and then find the corresponding angles $ACB$.From the calculations:

• $\sin(\angle ACB) = 0.851$
• The acute angle $ACB$ is approximately $58^\circ$
• The obtuse angle $ACB$ is approximately $122^\circ$

Thus, the two possible values for $\angle ACB$ are $58^\circ$ and $122^\circ$.

Part 2: Finding the obtuse angle $x$ in the lower triangle

For the lower triangle, we are given:

• $\angle A = 29^\circ$
• $BC = 6.6 \, \text{cm}$
• $AB = 4.3 \, \text{cm}$

We can again use the Sine Rule to calculate $x$. Let's apply the same process here:

$\frac{AB}{\sin(x)} = \frac{BC}{\sin(29^\circ)}$

Now, I will calculate $x$.For the second triangle:

• The value of $\sin(x) = 0.316$
• The corresponding angle $x$ is approximately $18^\circ$.

Summary of results:

1. The two possible values for $\angle ACB$ are:

   • $58^\circ$ (acute)
   • $122^\circ$ (obtuse)

2. The obtuse angle $x$ is approximately $18^\circ$.

Would you like more details on the steps or have any questions?

Here are 5 related questions to expand on this:

1. How does the sine rule apply to non-right-angled triangles?
2. Why are there two possible angles for $\angle ACB$ in the ambiguous case?
3. How can we verify which of the two solutions is more likely in a real-world scenario?
4. How do we determine whether the given angle is obtuse or acute?
5. Can the cosine rule be used for these triangles as well?

Tip: Always ensure that your calculator is set to the correct mode (degrees or radians) when solving trigonometric problems!