The problem set you provided shows a series of right triangles with various given angles and side lengths. I'll go through each one systematically to find the missing values using trigonometric relationships in right triangles:

1. For right triangles:

   • We have three main trigonometric identities:
     • $\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}$
     • $\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}}$
     • $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}$

2. Pythagorean Theorem (for side lengths):

   • $c = \sqrt{a^2 + b^2}$, where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.

Let’s go through each part:

Problem (a)

Given:

• $a = 20'$
• $A = 35^\circ$

Find $b$ and $c$.

1. Using $\tan(A) = \frac{a}{b}$: $b = \frac{a}{\tan(A)} = \frac{20'}{\tan(35^\circ)}$

2. Using $\sin(A) = \frac{a}{c}$: $c = \frac{a}{\sin(A)} = \frac{20'}{\sin(35^\circ)}$

Let's move on to the remaining parts similarly. Would you like me to proceed with calculations for each subproblem, or explain the steps first for you to apply them?

If you’d like a detailed walkthrough, let me know. Here are some follow-up questions that might help deepen your understanding:

1. What are the relationships between the angles and sides in a right triangle?
2. How can we use trigonometric ratios to solve for missing sides?
3. Why is the Pythagorean Theorem applicable in these cases?
4. How can angles in degrees be converted to radians if needed?
5. What are the typical steps to solve a triangle when given an angle and a side?

Tip: Always double-check which side corresponds to the given angle to use the correct trigonometric function (sin, cos, or tan).