1. **Stating the problem:** We have two parallel lines $m$ and $a$ cut by two transversals $r$ and $v$. Given angles are $5a$ near $r$ and $m$, $85^\circ$ near $r$ and $a$, and $15b+10$ near $v$ and $a$. We need to find relationships between $a$ and $b$. 2. **Using parallel lines and transversal angle rules:** When a transversal cuts parallel lines, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (sum to $180^\circ$). 3. **Relate $5a$ and $85^\circ$:** Since $5a$ is an angle on line $m$ with transversal $r$, and $85^\circ$ is the corresponding angle on line $a$ with the same transversal $r$, these are corresponding angles and must be equal: $$5a = 85$$ 4. **Solve for $a$:** $$a = \frac{85}{5} = 17$$ 5. **Relate $85^\circ$ and $15b+10$:** Angles $85^\circ$ (on line $a$ with transversal $r$) and $15b+10$ (on line $a$ with transversal $v$) are on the same line $a$ but different transversals. Since $r$ and $v$ are transversals, and lines $m$ and $a$ are parallel, angles on the same side of the transversal $v$ and line $a$ that are interior should be supplementary to the angle corresponding to $85^\circ$. Assuming $85^\circ$ and $15b+10$ are consecutive interior angles with respect to transversal $v$, they sum to $180^\circ$: $$85 + (15b + 10) = 180$$ 6. **Simplify and solve for $b$:** $$85 + 15b + 10 = 180$$ $$15b + 95 = 180$$ $$15b = 180 - 95$$ $$15b = 85$$ $$b = \frac{85}{15} = \frac{17}{3} \approx 5.67$$ **Final answers:** $$a = 17$$ $$b = \frac{17}{3}$$