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Let's break down the problem.

We are given the following information:

- Triangle \( QRS \) is similar to triangle \( FGH \), with the correspondence \( Q \leftrightarrow F \), \( R \leftrightarrow G \), and \( S \leftrightarrow H \).
- Each side of triangle \( FGH \) is \( \frac{1}{4} \) the length of the corresponding side in triangle \( QRS \).
- In triangle \( FGH \), the measures of the angles are:
  - \( \angle F = 35^\circ \)
  - \( \angle G = 8^\circ \)
  - \( FG = a \)
- In triangle \( QRS \), the measure of angle \( Q = b^\circ \) and \( QR = 20 \).

We need to find the value of \( a + b \).

### Step 1: Use Similarity of Triangles
Since \( \triangle QRS \sim \triangle FGH \), corresponding angles are equal. Therefore:
- \( \angle Q = \angle F = 35^\circ \), so \( b = 35 \).

### Step 2: Use the Sum of Angles in a Triangle
In triangle \( FGH \), the sum of the angles is:
\[
\angle F + \angle G + \angle H = 180^\circ
\]
Substituting the known values for \( \angle F \) and \( \angle G \):
\[
35^\circ + 8^\circ + \angle H = 180^\circ
\]
\[
\angle H = 180^\circ - 43^\circ = 137^\circ
\]
Thus, \( \angle H = 137^\circ \).

### Step 3: Solve for \( a \)
We know that the sides of \( \triangle FGH \) are \( \frac{1}{4} \) the corresponding sides of \( \triangle QRS \). Since \( QR \) in \( \triangle QRS \) corresponds to \( FG \) in \( \triangle FGH \) and \( QR = 20 \), we have:
\[
FG = \frac{1}{4} \times 20 = 5
\]
Thus, \( a = 5 \).

### Step 4: Compute \( a + b \)
Now we can find \( a + b \):
\[
a + b = 5 + 35 = 40
\]

### Final Answer:
The value of \( a + b \) is \( \boxed{40} \).

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions you can explore:
1. What is the relationship between the areas of two similar triangles?
2. How can the ratio of corresponding sides in similar triangles be used to find unknown lengths?
3. What are the different ways to prove triangles are similar?
4. How do the angle measures in a triangle determine the third angle?
5. What is the sum of the interior angles in any polygon?

**Tip**: In similar triangles, corresponding sides are proportional, and the ratios of corresponding sides are the same for all pairs of sides.

Triangle QRS is similar to triangle FGH such that Q, R, and S correspond to F, G, and H, respectively. Each side of triangle FGH has 1/4 the length of its corresponding side in triangle QRS. In triangle FGH, the measure of angle F is 35°, the measure of angle G is 8°, and FG = a. In triangle QRS, the measure of angle Q is b° and QR = 20. What is the value of a + b?

Solve a triangle similarity problem where triangle QRS is similar to triangle FGH. Learn to calculate corresponding angles and side lengths using proportions. This problem involves understanding the sum of angles in triangles and applying similarity ratios.

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