19 Mark for Review Triangle $$ Q R S $$ is similar to triangle $$ F G H $$ such that $$ Q $$, $$ R $$, and $$ S $$ correspond to $$ F, G $$ and $$ F $$. respectively. Each side of triangle $$ F G H $$ has $$ \frac{1}{4} $$ the length of its corresponding side in triangle $$ Q R S $$. In triangle $$ F G H $$, the measure of angle $$ F $$ is $$ 35^{\circ} $$ the measure of angle $$ Q $$ is $$ 8^{\circ} $$, and $$ F G=a $$. In friangle $$ Q R S $$, the measure of angle $$ Q $$ is $$ b^{\circ} $$ and $$ Q R=20 $$. What is the value of $$ a+b $$ ?

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To solve the problem, we will follow these steps:

1. **Identify the known values and relationships:**
   - Triangle $$ QRS $$ is similar to triangle $$ FGH $$.
   - The sides of triangle $$ FGH $$ are $$ \frac{1}{4} $$ the length of the corresponding sides in triangle $$ QRS $$.
   - The measure of angle $$ F $$ is $$ 35^\circ $$.
   - The measure of angle $$ Q $$ is $$ 8^\circ $$.
   - $$ FG = a $$ and $$ QR = 20 $$.
   - We need to find $$ a + b $$, where $$ b $$ is the measure of angle $$ QRS $$.

2. **Determine the measures of the angles in triangle $$ QRS $$:**
   - Since the triangles are similar, the corresponding angles are equal.
   - Therefore, angle $$ R $$ in triangle $$ FGH $$ corresponds to angle $$ S $$ in triangle $$ QRS $$.
   - The sum of angles in a triangle is $$ 180^\circ $$. Thus, we can find angle $$ R $$ in triangle $$ FGH $$:
     $$
     R = 180^\circ - F - G
     $$
   - We know $$ F = 35^\circ $$ and $$ G = 8^\circ $$:
     $$
     R = 180^\circ - 35^\circ - 8^\circ = 137^\circ
     $$

3. **Find the corresponding angle $$ S $$ in triangle $$ QRS $$:**
   - Since triangles $$ QRS $$ and $$ FGH $$ are similar, angle $$ S $$ corresponds to angle $$ R $$:
     $$
     S = 137^\circ
     $$

4. **Calculate the measure of angle $$ Q $$ in triangle $$ QRS $$:**
   - We already have $$ Q = 8^\circ $$ and $$ S = 137^\circ $$. We can find angle $$ R $$ in triangle $$ QRS $$:
     $$
     R = 180^\circ - Q - S = 180^\circ - 8^\circ - 137^\circ = 35^\circ
     $$

5. **Determine the lengths of the sides:**
   - Since $$ FG = a $$ and $$ QR = 20 $$, and knowing the ratio of the sides:
     $$
     \frac{FG}{QR} = \frac{1}{4}
     $$
   - Therefore:
     $$
     a = \frac{1}{4} \times 20 = 5
     $$

6. **Calculate $$ a + b $$:**
   - We have $$ a = 5 $$ and $$ b = 8 $$:
     $$
     a + b = 5 + 8 = 13
     $$

Thus, the final answer is $$ \boxed{13} $$.
$$ a + b = 13 $$

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