Given that lines $a$ and $b$ are parallel, and the measures of angles $3$ and $4$ are provided, we can solve for angle $8$ using geometric relationships. Here's a step-by-step process:

1. Identifying corresponding angles: Since $a$ and $b$ are parallel, angles $1$ and $5$, as well as angles $2$ and $6$, are corresponding angles and will have equal measures.

2. Using the fact that angles on a straight line sum to $180^\circ$:

   • Angles $3$ and $4$ are adjacent on the same straight line (they form a linear pair), so their sum must be $180^\circ$. Thus: $\text{Angle } 3 + \text{Angle } 4 = 180^\circ$ Substituting the values: $87^\circ + 54^\circ = 141^\circ$ The supplementary angle (Angle $1$) is: $180^\circ - 141^\circ = 39^\circ$

3. Corresponding angles principle: Since lines $a$ and $b$ are parallel, angle $1 =$ angle $5$. Therefore, angle $5 = 39^\circ$.

4. Angles on a straight line again: Now, angles $5$ and $8$ form a linear pair, so they must add up to $180^\circ$. Thus: $\text{Angle } 5 + \text{Angle } 8 = 180^\circ$ Substituting the value of angle $5$: $39^\circ + \text{Angle } 8 = 180^\circ$ Solving for angle $8$: $\text{Angle } 8 = 180^\circ - 39^\circ = 141^\circ$

Thus, the measure of angle $8$ is $141^\circ$.

Would you like more details or have any questions?

Here are 5 follow-up questions related to this:

1. What are corresponding angles in parallel lines?
2. Can you define the linear pair property in geometry?
3. How do alternate interior angles help in solving parallel line problems?
4. What is the sum of interior angles on the same side of a transversal?
5. How do supplementary angles relate to straight lines?

Tip: In any geometry problem involving parallel lines, always look for corresponding, alternate interior, and supplementary angles to find unknowns.