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Let's solve each of these problems step by step.

### Problem 40:
Given that \( \overrightarrow{BD} \) bisects \( \angle ABC \), the two angles on either side of the bisector must be equal:

\[
(4x - 2)^\circ = (3x + 18)^\circ
\]

Now, let's solve for \( x \):

\[
4x - 2 = 3x + 18
\]
Subtract \( 3x \) from both sides:
\[
x - 2 = 18
\]
Add 2 to both sides:
\[
x = 20
\]

Now that we know \( x = 20 \), substitute it back into one of the expressions for the angle:

\[
\angle ABD = 4x - 2 = 4(20) - 2 = 80 - 2 = 78^\circ
\]

Since \( \overrightarrow{BD} \) bisects \( \angle ABC \), the full angle \( \angle ABC \) is:

\[
m \angle ABC = 2 \times 78^\circ = 156^\circ
\]

---

### Problem 41:
Here, \( \overrightarrow{BD} \) bisects \( \angle ABC \), so:

\[
(2x + 20)^\circ = 4x^\circ
\]

Now solve for \( x \):

\[
2x + 20 = 4x
\]
Subtract \( 2x \) from both sides:
\[
20 = 2x
\]
Divide both sides by 2:
\[
x = 10
\]

Now substitute \( x = 10 \) back into the expression:

\[
\angle ABD = 2x + 20 = 2(10) + 20 = 20 + 20 = 40^\circ
\]

Since \( \overrightarrow{BD} \) bisects \( \angle ABC \), the full angle \( \angle ABC \) is:

\[
m \angle ABC = 2 \times 40^\circ = 80^\circ
\]

---

### Problem 42:
In this case, \( \overrightarrow{BD} \) bisects \( \angle ABC \), so:

\[
\left( \frac{x}{2} + 17 \right)^\circ = (x - 33)^\circ
\]

Now solve for \( x \):

\[
\frac{x}{2} + 17 = x - 33
\]
Subtract \( \frac{x}{2} \) from both sides:
\[
17 = \frac{x}{2} - 33
\]
Add 33 to both sides:
\[
50 = \frac{x}{2}
\]
Multiply both sides by 2:
\[
x = 100
\]

Now substitute \( x = 100 \) back into the expression:

\[
\angle ABD = \frac{x}{2} + 17 = \frac{100}{2} + 17 = 50 + 17 = 67^\circ
\]

Since \( \overrightarrow{BD} \) bisects \( \angle ABC \), the full angle \( \angle ABC \) is:

\[
m \angle ABC = 2 \times 67^\circ = 134^\circ
\]

---

### Summary of Results:
- **Problem 40**: \( m \angle ABC = 156^\circ \)
- **Problem 41**: \( m \angle ABC = 80^\circ \)
- **Problem 42**: \( m \angle ABC = 134^\circ \)

Would you like any further details or explanations?

---

Here are 5 related questions to expand your understanding:
1. What is the geometric significance of an angle bisector in a triangle?
2. How can you verify the accuracy of angle bisectors in geometrical constructions?
3. Can you think of real-life applications where angle bisectors play a role?
4. How would the angle values change if the diagram were scaled up or down?
5. What would happen to the angles if the bisector did not perfectly bisect the main angle?

**Tip:** When solving algebraic equations involving angles, always check if the angle measures add up to the expected total (e.g., 180° in a triangle).

In each diagram, BD bisects ∠ABC. Find m∠ABC.

Learn how to find the angle ∠ABC using algebraic expressions when given that BD bisects the angle. This problem involves using linear equations and the angle bisector theorem. Suitable for students in Grades 8-10.

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