To solve for the angles $\angle ABD$, $\angle CBD$, and $\angle ABC$, we can use the information provided in the diagram and the fact that $BD$ bisects $\angle ABC$.

Given:

• $BD$ bisects $\angle ABC$, meaning that $m\angle ABD = m\angle CBD$.
• $m\angle ABD = (3x + 6)^\circ$
• $m\angle CBD = (7x - 18)^\circ$

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

$3x + 6 = 7x - 18$

Step 1: Solve for $x$

$3x + 6 = 7x - 18$ Subtract $3x$ from both sides: $6 = 4x - 18$ Add 18 to both sides: $24 = 4x$ Divide both sides by 4: $x = 6$

Step 2: Find $m\angle ABD$ and $m\angle CBD$

Substitute $x = 6$ into the expressions for $\angle ABD$ and $\angle CBD$:

$m\angle ABD = 3(6) + 6 = 18 + 6 = 24^\circ$ $m\angle CBD = 7(6) - 18 = 42 - 18 = 24^\circ$

Step 3: Find $m\angle ABC$

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

$m\angle ABC = m\angle ABD + m\angle CBD = 24^\circ + 24^\circ = 48^\circ$

Final Answers:

$m\angle ABD = 24^\circ$ $m\angle CBD = 24^\circ$ $m\angle ABC = 48^\circ$

Would you like further details on this, or do you have any questions?

Here are some related questions you might find helpful:

1. How would the solution change if $BD$ did not bisect $\angle ABC$?
2. What would happen if the expression for $\angle ABD$ involved quadratic terms?
3. How can you verify that the angles in a bisected angle are equal?
4. Can you use trigonometric methods to find the angles in different ways?
5. How would you approach the problem if the diagram was scaled or rotated?

Tip: Always verify whether the bisector splits the angle into equal parts to apply symmetry and simplify the problem.