1. **State the problem:** We need to find the measure of angle $\angle AOB$ given that $m\angle AOB = 4y - 3$ and $m\angle BOC = 6y - 17$. We are told that $\angle AOB$ and $\angle BOC$ are complementary angles. 2. **Recall the complementary angle rule:** Complementary angles add up to $90^\circ$. So, $$m\angle AOB + m\angle BOC = 90$$ 3. **Write the equation using the given expressions:** $$ (4y - 3) + (6y - 17) = 90 $$ 4. **Simplify the equation:** $$ 4y - 3 + 6y - 17 = 90 $$ $$ 10y - 20 = 90 $$ 5. **Solve for $y$:** Add 20 to both sides: $$ 10y - \cancel{20} + \cancel{20} = 90 + 20 $$ $$ 10y = 110 $$ Divide both sides by 10: $$ \frac{10y}{\cancel{10}} = \frac{110}{\cancel{10}} $$ $$ y = 11 $$ 6. **Find $m\angle AOB$ by substituting $y=11$ into $4y - 3$:** $$ m\angle AOB = 4(11) - 3 = 44 - 3 = 41 $$ **Final answer:** $$ m\angle AOB = 41^\circ $$