https://file.aiquickdraw.com/mathbot/file/1725597028736rp864580.jpg

Let's solve the given problems from the image for problems 28 and 30.

### Problem 28:
\[
m\angle ABC = 4x - 12, \quad m\angle ABD = 24
\]
Since \( BD \) bisects \( \angle ABC \), we know:
\[
m\angle ABD = m\angle DBC
\]
This implies:
\[
m\angle ABC = 2 \times m\angle ABD
\]
Thus, substitute the given value:
\[
4x - 12 = 2 \times 24
\]
Simplify:
\[
4x - 12 = 48
\]
Add 12 to both sides:
\[
4x = 60
\]
Divide by 4:
\[
x = 15
\]
Now substitute \( x = 15 \) back into the expression for \( m\angle ABC \):
\[
m\angle ABC = 4(15) - 12 = 60 - 12 = 48
\]
So:
\[
m\angle ABC = 48^\circ
\]

---

### Problem 30:
\[
m\angle ABD = 3x + 20, \quad m\angle CBD = 6x - 16
\]
Since \( BD \) bisects \( \angle ABC \), we know:
\[
m\angle ABD = m\angle DBC
\]
So, equate the two expressions:
\[
3x + 20 = 6x - 16
\]
Solve for \( x \):
\[
20 + 16 = 6x - 3x
\]
\[
36 = 3x
\]
\[
x = 12
\]
Now substitute \( x = 12 \) back into the expressions for \( m\angle ABD \) and \( m\angle DBC \):
\[
m\angle ABD = 3(12) + 20 = 36 + 20 = 56^\circ
\]
\[
m\angle ABC = 2 \times m\angle ABD = 2 \times 56 = 112^\circ
\]

---

### Summary of Results:
- **Problem 28**: \( m\angle ABC = 48^\circ \)
- **Problem 30**: \( m\angle ABC = 112^\circ \)

Let me know if you'd like further details or explanations! 

Here are 5 related questions:
1. How do you verify if an angle bisector equally divides an angle?
2. What are other geometric properties of angle bisectors in a triangle?
3. How can algebraic techniques simplify angle-related problems?
4. What happens when two angles are supplementary in a bisected configuration?
5. Can you generalize this process for solving any linear equation related to angles?

**Tip**: Always substitute back the value of \(x\) to check the correctness of your final angle calculations!

Learn how to solve problems involving angle bisectors with a detailed step-by-step solution for problems 28 and 30. This problem covers key algebraic concepts, including solving linear equations and applying the angle bisector theorem. Suitable for students in Grades 9-11.

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