To solve each of these triangles, I'll proceed based on the given information using trigonometric relationships. I’ll go step-by-step for each item:

1) Given: $a = 8 \, \text{cm}$, $b = 15 \, \text{cm}$

Since two sides are given, we can use the Pythagorean theorem or trigonometric rules (if a right triangle is assumed) or use the Law of Cosines if the angle between them is unknown.

Solution:

1. Calculate $c$ (assuming it's a right triangle): $c = \sqrt{a^2 + b^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \, \text{cm}$

2) Given: $B = 54^\circ$, $c = 20 \, \text{yd}$

With one angle and one side opposite that angle, we can use the Law of Sines to find the other sides if more information on other angles or sides is provided.

3) Given: $b = 10 \, \text{ft}$, $c = 14 \, \text{ft}$

Assuming this is a right triangle, we can calculate $a$ using the Pythagorean theorem.

Solution:

$a = \sqrt{c^2 - b^2} = \sqrt{14^2 - 10^2} = \sqrt{196 - 100} = \sqrt{96} \approx 9.80 \, \text{ft}$

4) Given: $A = 13^\circ$, $b = 15 \, \text{cm}$

With angle $A$ and side $b$, we may use the Law of Sines if the opposite side or another angle is given.

5) Given: $a = 13 \, \text{in}$, $c = 20 \, \text{in}$

If $c$ is the hypotenuse in a right triangle, we can use the Pythagorean theorem to solve for $b$.

Solution:

$b = \sqrt{c^2 - a^2} = \sqrt{20^2 - 13^2} = \sqrt{400 - 169} = \sqrt{231} \approx 15.20 \, \text{in}$

6) Given: $B = 45^\circ$, $b = 90 \, \text{m}$

With angle $B$ and side $b$, we can find the other sides if more information about angle relationships or side proportions is available.

Would you like further details on any specific solution or help with a particular part of this problem set?

Here are some additional questions to consider:

1. How does the Law of Sines work for non-right triangles?
2. What alternative methods exist for solving triangles if only angles are given?
3. How do trigonometric identities assist in triangle solving?
4. Can the Law of Cosines be applied for all types of triangles?
5. How can these methods be used in real-world applications, such as navigation?

Tip: For ambiguous cases in triangle solving, check if the given information is sufficient to ensure a unique solution.