The problem asks to solve for $x$ and find the measure of $\angle ABC$, given that $BD$ bisects $\angle ABC$, meaning $\angle ABD = \angle CBD$. The expressions for the two angles are:

• $\angle ABD = 11x - 4$
• $\angle CBD = 8x + 5$

Since $BD$ bisects $\angle ABC$, we can set the two expressions equal to each other:

$11x - 4 = 8x + 5$

Step 1: Solve for $x$

1. Subtract $8x$ from both sides: $11x - 8x - 4 = 5$ $3x - 4 = 5$

2. Add 4 to both sides: $3x = 9$

3. Divide by 3: $x = 3$

Step 2: Find $\angle ABC$

Since $BD$ bisects $\angle ABC$, we know that $\angle ABC = 2 \times \angle ABD$ (or $\angle ABC = 2 \times \angle CBD$, as they are equal).

First, substitute $x = 3$ into either expression for $\angle ABD$ or $\angle CBD$. Let's use $\angle ABD = 11x - 4$:

$\angle ABD = 11(3) - 4 = 33 - 4 = 29^\circ$

Since $\angle ABD = \angle CBD$, $\angle ABC = 2 \times 29 = 58^\circ$.

Final Answer:

• $x = 3$
• $\angle ABC = 58^\circ$

Would you like further details on any of these steps or clarifications?

Here are 5 related questions to expand your understanding:

1. What happens if $BD$ does not bisect $\angle ABC$?
2. How can we use properties of angle bisectors in triangles?
3. How do we solve if $\angle ABD$ and $\angle CBD$ were unequal?
4. What other methods can be used to verify the measure of an angle bisector?
5. How does this problem relate to solving for unknowns in geometry?

Tip: When solving for variables in angle relationships, always use given geometrical properties, like bisectors, as they offer critical equalities or proportions.