Let's analyze and solve the questions step-by-step.

1. Triangle solvable using the Law of Cosines

The Law of Cosines is used when:

• Two sides and the included angle are known (SAS case), or
• All three sides are known (SSS case).

Given options:

• A: $\angle B = 32^\circ, \angle C = 118^\circ, c = 42$ → Known: two angles and one side → Use the Law of Sines instead.
• B: $\angle B = 82^\circ, \angle C = 75^\circ, c = 24$ → Known: two angles and one side → Use the Law of Sines.
• C: $\angle A = 120^\circ, a = 10, b = 19$ → Known: two sides $a, b$ and included angle $\angle A$ → Use the Law of Cosines.
• D: $\angle A = 130^\circ, b = 5, c = 18$ → Known: two sides and an included angle → Law of Cosines can be used.

Answer: Both options C and D can use the Law of Cosines. However, if the question intends for just one, then likely C is the best choice as it provides clear SAS format.

2. Measure of the smallest angle

The sides of a triangle are $27, 32, 34$. To find the smallest angle:

Use the Law of Cosines: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$ Here, the smallest angle is opposite the smallest side ($a = 27$).

Substitute:

• $a = 27, b = 32, c = 34$ $\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{32^2 + 34^2 - 27^2}{2(32)(34)}.$

Calculate step-by-step:

• $32^2 = 1024, \, 34^2 = 1156, \, 27^2 = 729$
• $b^2 + c^2 - a^2 = 1024 + 1156 - 729 = 1451$
• $2bc = 2(32)(34) = 2176$

$\cos A = \frac{1451}{2176} \approx 0.667$

Find $A$ using $\cos^{-1}$: $A = \cos^{-1}(0.667) \approx 48^\circ.$

Answer: C. 48

3. What is an oblique triangle?

An oblique triangle is any triangle that does not contain a right angle. It can be:

• Acute (all angles $< 90^\circ$) or
• Obtuse (one angle $> 90^\circ$).

Summary of Answers:

1. C: $\angle A = 120^\circ, a = 10, b = 19$
2. C: 48°
3. Oblique Triangle: It is a triangle with no right angle.

Do you want me to explain any part further? Here are 5 related questions:

1. What is the formula for the Law of Cosines, and when do we use it?
2. How can you determine the largest angle in a triangle given three sides?
3. What is the Law of Sines, and how does it differ from the Law of Cosines?
4. How do you classify triangles based on their angles and sides?
5. How can you verify if a triangle is obtuse using its side lengths?

Tip: Always identify the known parts of a triangle (angles and sides) to determine whether to use the Law of Sines or Cosines.