Question: 23. The measure of $\angle ABD$ and $\angle DBC$ given $m\angle ABC = 77^\circ$ with angles $x - 2$ and $3x + 2$. 24. Find the measure of $\angle ABD$ and $\angle DBC$ given $m\angle ABC = 149^\circ$ with angles $4x - 8$ and $3x + 8$.

1. **Problem 23:** Given $m\angle ABC = 77^\circ$, find $\angle ABD = x - 2$ and $\angle DBC = 3x + 2$. 2. Since $\angle ABD$ and $\angle DBC$ are adjacent and form $\angle ABC$, their measures add up: $$ (x - 2) + (3x + 2) = 77 $$ 3. Simplify the equation: $$ x - 2 + 3x + 2 = 77 $$ $$ 4x = 77 $$ 4. Solve for $x$: $$ x = \frac{77}{4} $$ 5. Calculate each angle: $$ \angle ABD = x - 2 = \frac{77}{4} - 2 = \frac{77}{4} - \frac{8}{4} = \frac{69}{4} = 17.25^\circ $$ $$ \angle DBC = 3x + 2 = 3 \times \frac{77}{4} + 2 = \frac{231}{4} + \frac{8}{4} = \frac{239}{4} = 59.75^\circ $$ --- 6. **Problem 24:** Given $m\angle ABC = 149^\circ$, find $\angle ABD = 4x - 8$ and $\angle DBC = 3x + 8$. 7. Set up the equation: $$ (4x - 8) + (3x + 8) = 149 $$ 8. Simplify: $$ 4x - 8 + 3x + 8 = 149 $$ $$ 7x = 149 $$ 9. Solve for $x$: $$ x = \frac{149}{7} $$ 10. Calculate each angle: $$ \angle ABD = 4x - 8 = 4 \times \frac{149}{7} - 8 = \frac{596}{7} - \frac{56}{7} = \frac{540}{7} \approx 77.14^\circ $$ $$ \angle DBC = 3x + 8 = 3 \times \frac{149}{7} + 8 = \frac{447}{7} + \frac{56}{7} = \frac{503}{7} \approx 71.86^\circ $$ --- **Final answers:** - For problem 23: $\angle ABD = 17.25^\circ$, $\angle DBC = 59.75^\circ$ - For problem 24: $\angle ABD \approx 77.14^\circ$, $\angle DBC \approx 71.86^\circ$