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### Problem 8:
**The larger of two supplementary angles exceeds the smaller by 4°. Find the measure of the larger angle.**

- Let the smaller angle be \( x \).
- Since the angles are supplementary, their sum is 180°, so the larger angle is \( 180^\circ - x \).
- The problem states that the larger angle exceeds the smaller by 4°, so:
  \[
  (180^\circ - x) = x + 4^\circ
  \]

- Solve for \( x \):
  \[
  180^\circ - x = x + 4^\circ
  \]
  \[
  180^\circ - 4^\circ = 2x
  \]
  \[
  176^\circ = 2x
  \]
  \[
  x = 88^\circ
  \]

- The smaller angle is \( 88^\circ \), so the larger angle is \( 88^\circ + 4^\circ = 92^\circ \).

Thus, the larger angle is **92°**.

---

### Problem 9:
**One of two complementary angles added to one-half the other yields 72°. Find half the measure of the larger.**

- Let the two complementary angles be \( x \) and \( y \), so:
  \[
  x + y = 90^\circ
  \]
- The problem states that one angle added to half the other equals 72°:
  \[
  x + \frac{y}{2} = 72^\circ
  \]
  
  Substitute \( y = 90^\circ - x \) into the equation:
  \[
  x + \frac{90^\circ - x}{2} = 72^\circ
  \]
  \[
  x + \frac{90^\circ}{2} - \frac{x}{2} = 72^\circ
  \]
  \[
  x + 45^\circ - \frac{x}{2} = 72^\circ
  \]
  \[
  \frac{2x}{2} - \frac{x}{2} = 72^\circ - 45^\circ
  \]
  \[
  \frac{x}{2} = 27^\circ
  \]
  \[
  x = 54^\circ
  \]

- Since \( y = 90^\circ - x \), we find that \( y = 36^\circ \).

- Half of the larger angle (\( x = 54^\circ \)) is:
  \[
  \frac{54^\circ}{2} = 27^\circ
  \]

Thus, half the measure of the larger angle is **27°**.

---

### Problem 10:
**Given: \( XY \perp YW \), \( AB \perp BC \). Find \( \angle DBC \).**

In the diagram, it appears that you are dealing with two perpendicular lines \( XY \) and \( YW \), as well as \( AB \perp BC \). This setup typically forms right angles and can involve concepts from coordinate geometry or triangle properties. Without further information from the image, such as angle labels or more details, it is difficult to provide a precise solution. If you could clarify the relationships between the points or any known angles, that would help in solving the problem.

---

Would you like further details on any of these problems?

Here are five related questions that might interest you:
1. How do you solve for angles when two lines are perpendicular?
2. What is the difference between supplementary and complementary angles?
3. How can you find the measure of unknown angles using algebra?
4. How do perpendicularity conditions affect angle measures in geometry?
5. How can the Pythagorean theorem be applied in problems with perpendicular lines?

**Tip**: In angle-related problems, always check if the angles are complementary or supplementary to simplify the equation setup.

The larger of two supplementary angles exceeds the smaller by 4°. Find the measure of the larger angle. One of two complementary angles added to one-half the other yields 72°. Find half the measure of the larger. Given: XY ⊥ YW, AB ⊥ BC. Find m∠DBC.

This page explains how to find the larger of two supplementary angles that exceeds the smaller by 4°, and how to find half the measure of a larger complementary angle given one-half the other. It also provides an introduction to solving for unknown angles in geometric proofs involving perpendicular lines.

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