I see the uploaded image contains several math problems related to geometry and solving for variables. Let’s address them one by one:

Problem 8: Determine if $ST$ is parallel to $PR$.

To determine if $ST$ is parallel to $PR$, we need to check if the triangle adheres to the Triangle Proportionality Theorem, which states that if a line parallel to one side of a triangle divides the other two sides proportionally, then the line is parallel to the third side.

The triangle dimensions are:

• $SP = 7, PT = 10, PR = 16$.

Using proportions: $\frac{SP}{PT} = \frac{7}{10}, \quad \frac{ST}{PR} = \frac{11.2}{16}.$

Check: $\frac{7}{10} = 0.7, \quad \frac{11.2}{16} = 0.7.$ Since these proportions are equal, $ST$ is parallel to $PR$.

Problem 10: Determine if $ST$ is parallel to $PR$.

Here the triangle dimensions are:

• $SP = 19, PT = 38, PR = 52$.

Using proportions: $\frac{SP}{PT} = \frac{19}{38} = 0.5, \quad \frac{ST}{PR} = \frac{2A}{52}.$

We need $2A$ to satisfy $\frac{2A}{52} = 0.5$: $2A = 0.5 \times 52 = 26 \quad \text{so } A = 13.$

Thus, if $A = 13$, $ST$ is parallel to $PR$.

Problem 11: Solve for $x$.

The given diagram shows a transversal cutting two parallel lines, forming pairs of proportional segments. By the property of transversals: $\frac{25}{30} = \frac{48}{x}.$

Cross-multiply: $25x = 30 \times 48.$

Simplify: $25x = 1440 \quad \Rightarrow \quad x = \frac{1440}{25} = 57.6.$

So, $x = 57.6$.

Problem 14: Solve for $x$.

This involves solving for $x$ using the given segments of a triangle divided by a transversal.

The proportions are: $\frac{x-3}{21} = \frac{x-1}{27}.$

Cross-multiply: $27(x - 3) = 21(x - 1).$

Expand: $27x - 81 = 21x - 21.$

Simplify: $6x = 60 \quad \Rightarrow \quad x = 10.$

So, $x = 10$.

Problem 15: Solve for $x$.

Here we equate the ratios of the parallel lines: $\frac{7x - 11}{20} = \frac{4x - 2}{35}.$

Cross-multiply: $35(7x - 11) = 20(4x - 2).$

Expand: $245x - 385 = 80x - 40.$

Simplify: $165x = 345 \quad \Rightarrow \quad x = \frac{345}{165} = 2.09.$

So, $x \approx 2.09$.

Would you like detailed explanations for any of these?
Here are five related questions to extend this:

1. How is the Triangle Proportionality Theorem applied in different scenarios?
2. What happens if proportions are not equal when solving for parallel lines?
3. How does a transversal affect angle relationships in parallel lines?
4. Can the proportionality rule be applied in non-triangular figures?
5. What is the difference between corresponding and alternate angles in parallel lines?

Tip: Always double-check your setup of proportions to avoid simple errors in cross-multiplication!