We are given that $BD$ bisects $\angle ABC$, meaning that $\angle ABD = \angle CBD$. Also, we have the expressions for the two angles:

$\angle ABD = 8x + 35^\circ \quad \text{and} \quad \angle CBD = 11x + 23^\circ.$

Since $BD$ bisects $\angle ABC$, we have:

$8x + 35^\circ = 11x + 23^\circ.$

Let's solve for $x$:

$8x + 35 = 11x + 23.$

Subtract $8x$ from both sides:

$35 = 3x + 23.$

Subtract 23 from both sides:

$12 = 3x.$

Now, divide by 3:

$x = 4.$

Now that we have $x = 4$, we can substitute this value back into the expressions for $\angle ABD$ and $\angle CBD$.

1. $\angle ABD = 8(4) + 35 = 32 + 35 = 67^\circ$.
2. $\angle CBD = 11(4) + 23 = 44 + 23 = 67^\circ$.

Since $BD$ bisects $\angle ABC$, the full angle $\angle ABC$ is the sum of $\angle ABD$ and $\angle CBD$:

$\angle ABC = 67^\circ + 67^\circ = 134^\circ.$

Final answers:

• $m\angle ABD = 67^\circ$
• $m\angle CBD = 67^\circ$
• $m\angle ABC = 134^\circ$

Would you like a detailed breakdown of the steps? Here are some additional questions to consider:

1. How do we verify that $BD$ bisects $\angle ABC$?
2. What are the implications of angle bisectors in triangles?
3. How can we apply this method to different geometric problems involving bisectors?
4. What if $BD$ did not bisect $\angle ABC$, how would the solution change?
5. How do we find the external angles related to this problem?

Tip: In problems involving bisectors, always remember that the bisector divides an angle into two equal parts, which can simplify solving for unknowns.