We are given that $BD$ bisects $\angle ABC$, and the measurements of the angles are expressed in terms of $x$. Here's how we can solve for $m\angle ABD$, $m\angle CBD$, and $m\angle ABC$:

Step-by-Step Solution:

1. Expression for the angles:

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

2. Angle bisector property: Since $BD$ bisects $\angle ABC$, we know that: $m\angle ABD = m\angle CBD$

   Therefore, we can set the two expressions equal to each other: $3x + 6 = 7x - 18$

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

4. Find $m\angle ABD$ and $m\angle CBD$: Substitute $x = 6$ into the expressions for $m\angle ABD$ and $m\angle CBD$: $m\angle ABD = 3(6) + 6 = 18 + 6 = 24^\circ$ $m\angle CBD = 7(6) - 18 = 42 - 18 = 24^\circ$

5. Find $m\angle ABC$: Since $BD$ bisects $\angle ABC$, we know: $m\angle ABC = m\angle ABD + m\angle CBD = 24^\circ + 24^\circ = 48^\circ$

Final Answers:

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

Would you like further details or have any other questions?

Here are 5 related questions:

1. How would the solution change if $BD$ did not bisect $\angle ABC$?
2. What happens if $BD$ bisects a right angle at $\angle ABC$?
3. Can angle bisectors always divide an angle into two equal parts in non-Euclidean geometry?
4. How would you find $x$ if the bisected angle had been expressed differently?
5. How does the concept of angle bisectors apply in triangle constructions?

Tip: When solving angle-related problems, always check if there are additional geometric properties, such as symmetry or bisectors, to simplify the solution.