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The image shows a problem involving angles formed by a transversal intersecting two parallel lines. The task is to find the measure of ∠NPO.

The angles in the diagram are expressed algebraically:
- ∠KPI = \( (61 - x)^\circ \)
- ∠JPO = \( (75 - 3x)^\circ \)

Since lines are parallel, you can use the properties of corresponding, alternate interior, and supplementary angles to solve for \( x \).

### Steps to Solve:

1. **Identify corresponding angles**: 
   ∠KPI and ∠JPO are vertical angles, meaning they are equal. Therefore, you can set up the equation:
   \[
   61 - x = 75 - 3x
   \]

2. **Solve for \( x \)**:
   Rearranging the equation:
   \[
   61 - 75 = -3x + x
   \]
   \[
   -14 = -2x
   \]
   Dividing by -2:
   \[
   x = 7
   \]

3. **Substitute \( x = 7 \) back into the expression for ∠NPO**:
   ∠NPO is equal to ∠JPO since they are corresponding angles. So, we substitute \( x = 7 \) into the expression for ∠JPO:
   \[
   75 - 3(7) = 75 - 21 = 54^\circ
   \]

Thus, the measure of ∠NPO is **54°**.

Would you like more details or have any further questions?

Here are some related questions to deepen understanding:
1. How are alternate interior angles used in transversals?
2. Why are vertical angles always equal?
3. How do you identify supplementary angles in a transversal problem?
4. Can corresponding angles be supplementary under certain conditions?
5. What other geometric properties can be applied to transversals?

**Tip**: Always look for angle pairs such as corresponding or alternate interior angles when dealing with parallel lines and transversals.

Find the measure of ∠NPO in the given diagram, where the angles are algebraically represented as (61 - x)° and (75 - 3x)°.

Learn how to find the measure of ∠NPO using algebraic expressions and angle relationships in a problem involving a transversal and parallel lines. This geometry problem is ideal for students in Grades 8-10 and covers key concepts like corresponding angles and vertical angles.

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