1. **Problem Statement:** Given two parallel lines $a \parallel b$ and angles with measures $m\angle 2 = 2x$, $m\angle 3 = 3x - 10$, and $m\angle 4 = 5x - 10$, find the measures of all numbered angles. 2. **Key Concept:** When two lines are parallel, alternate interior angles and corresponding angles are equal. Also, angles around a point sum to $360^\circ$, and angles on a straight line sum to $180^\circ$. 3. **Step 1: Use the relationship between angles 2, 3, and 4.** Since angles 2, 3, and 4 are adjacent around the intersection on line $a$, and assuming angles 2 and 4 are on a straight line with angle 3 between them, we can write: $$m\angle 2 + m\angle 3 + m\angle 4 = 180^\circ$$ Substitute the expressions: $$2x + (3x - 10) + (5x - 10) = 180$$ 4. **Step 2: Simplify the equation:** $$2x + 3x - 10 + 5x - 10 = 180$$ $$ (2x + 3x + 5x) - 20 = 180$$ $$10x - 20 = 180$$ 5. **Step 3: Solve for $x$:** $$10x = 180 + 20$$ $$10x = 200$$ $$x = \frac{200}{10} = 20$$ 6. **Step 4: Find the measures of angles 2, 3, and 4:** $$m\angle 2 = 2x = 2 \times 20 = 40^\circ$$ $$m\angle 3 = 3x - 10 = 3 \times 20 - 10 = 60 - 10 = 50^\circ$$ $$m\angle 4 = 5x - 10 = 5 \times 20 - 10 = 100 - 10 = 90^\circ$$ 7. **Step 5: Find the other angles using parallel line properties:** - Angle 1 is vertically opposite to angle 2, so $m\angle 1 = 40^\circ$. - Angle 5 is corresponding to angle 1 (since $a \parallel b$), so $m\angle 5 = 40^\circ$. - Angle 6 is vertically opposite to angle 5, so $m\angle 6 = 40^\circ$. - Angle 7 and angle 6 form a linear pair on line $b$, so: $$m\angle 7 = 180^\circ - m\angle 6 = 180 - 40 = 140^\circ$$ - Angle 8 is vertically opposite to angle 7, so $m\angle 8 = 140^\circ$. **Final answers:** $$m\angle 1 = 40^\circ, m\angle 2 = 40^\circ, m\angle 3 = 50^\circ, m\angle 4 = 90^\circ, m\angle 5 = 40^\circ, m\angle 6 = 40^\circ, m\angle 7 = 140^\circ, m\angle 8 = 140^\circ$$