To solve the given problem, we have the following information:

• $BD$ bisects $\angle ABC$, meaning $m\angle ABD = m\angle CBD$.
• $m\angle ABC = (2 - 16x)^\circ$.
• $m\angle ABD = (2x + 81)^\circ$.

Since $BD$ bisects $\angle ABC$, the two angles $\angle ABD$ and $\angle CBD$ are equal, and their sum equals $m\angle ABC$.

Step-by-step approach:

1. Set up the equation: $m\angle ABC = m\angle ABD + m\angle CBD$ Since $m\angle ABD = m\angle CBD$, we can write: $m\angle ABC = 2 \times m\angle ABD$ Substituting the expressions for $m\angle ABC$ and $m\angle ABD$: $(2 - 16x) = 2 \times (2x + 81)$

2. Solve the equation: $(2 - 16x) = 2(2x + 81)$ Simplifying the right side: $2 - 16x = 4x + 162$ Move all terms involving $x$ to one side and constants to the other: $-16x - 4x = 162 - 2$ $-20x = 160$ Solve for $x$: $x = -8$

3. Find the angle measures:

   • $m\angle ABD = 2x + 81 = 2(-8) + 81 = -16 + 81 = 65^\circ$
   • $m\angle CBD = m\angle ABD = 65^\circ$
   • $m\angle ABC = 2 - 16x = 2 - 16(-8) = 2 + 128 = 130^\circ$

Final answers:

• $m\angle ABD = 65^\circ$
• $m\angle CBD = 65^\circ$
• $m\angle ABC = 130^\circ$

Let me know if you would like more details or have any questions!

Here are 5 related questions to expand on the concept:

1. How do we know that the angles $\angle ABD$ and $\angle CBD$ are equal?
2. What is the geometric property of an angle bisector in a triangle?
3. How would the solution change if $BD$ was not a bisector?
4. What happens if the sum of $\angle ABC$ is greater than 180° in a triangle?
5. Can you find the exterior angle formed by extending $AB$?

Tip: When working with angle bisectors, always remember that it divides the angle into two equal parts!