Let's solve this step-by-step.

Step 1: Understanding Given Information

For triangle $ABC$:

• $\angle A = 30^\circ$
• $a = 10$ units (side opposite angle $A$)
• $c = 12$ units (side opposite angle $C$)
• Side $b$ is unknown but said to be equal to the sum of $\angle A$ and $\angle C$ in degrees.

We will:

1. Use the sine rule to find $\angle B$.
2. Verify if triangle $ABC$ satisfies the triangle inequality.
3. If valid, find the length of $b$ and then $\angle C$ using the sine rule.

Step 2: Using the Sine Rule to Find $\angle B$

The sine rule states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Let's first use $\frac{a}{\sin A} = \frac{c}{\sin C}$ to find $\angle C$ because $b$ depends on the sum of $\angle A$ and $\angle C$ in degrees.

1. Setting up the equation: $\frac{10}{\sin 30^\circ} = \frac{12}{\sin C}$

   Since $\sin 30^\circ = 0.5$, this simplifies to: $\frac{10}{0.5} = \frac{12}{\sin C}$ $20 = \frac{12}{\sin C}$ $\sin C = \frac{12}{20} = 0.6$

2. Finding $\angle C$: $C = \sin^{-1}(0.6) \approx 36.87^\circ$

Step 3: Calculating $b$

Since $b$ is said to be equal to the sum of $\angle A$ and $\angle C$ in degrees: $b = 30^\circ + 36.87^\circ = 66.87^\circ$

Step 4: Using the Sine Rule to Find $\angle B$

Now, we know:

• $\angle A = 30^\circ$
• $\angle C \approx 36.87^\circ$
• $\angle B = 180^\circ - 30^\circ - 36.87^\circ$

Calculating $\angle B$: $B \approx 180^\circ - 66.87^\circ = 113.13^\circ$

Step 5: Verify the Triangle Inequality

For triangle inequality to hold:

• $a + b > c$
• $a + c > b$
• $b + c > a$

Substituting:

1. $10 + 66.87 > 12$ — True
2. $10 + 12 > 66.87$ — False (This inequality does not hold)

Since the triangle inequality is not satisfied, triangle $ABC$ is not valid with the given values of sides and angles.

If you have any further questions or would like more details, please let me know!

Additional Questions

1. How would the solution change if side $b$ were defined differently?
2. Can the sine rule be applied if only two angles are known?
3. How can we determine if a triangle is valid without calculating side lengths?
4. What is the significance of the triangle inequality in geometry?
5. How would the solution proceed if the triangle were valid?

Tip

When solving for unknown angles or sides in triangles, always check if the solution satisfies the triangle inequality to confirm validity.