Triangle N P O is shown. Line O N extends through point M to form exterior angle P N M. Angle N P O is 38 degrees. Angle P O N is 39 degrees. Exterior angle P N M is x degrees.
Which statement about the value of x is true?

x > 38
x < 39
x < 77
x  > 103

To solve this problem, we'll use the **exterior angle theorem**. This theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Given:
- \(\angle NPO = 38^\circ\)
- \(\angle PON = 39^\circ\)
- \(\angle PNM = x^\circ\) (exterior angle)

Using the exterior angle theorem:
\[
x = \angle NPO + \angle PON
\]
Substituting the given values:
\[
x = 38^\circ + 39^\circ = 77^\circ
\]

So, \(x = 77^\circ\).

Now, let's analyze the statements:

1. \(x > 38\) — True, because \(77 > 38\).
2. \(x < 39\) — False, because \(77 > 39\).
3. \(x < 77\) — False, because \(x = 77\).
4. \(x > 103\) — False, because \(77 < 103\).

The correct statement is **\(x > 38\)**.

Would you like more details or have any questions? Here are some related questions to consider:

1. How does the exterior angle theorem help in triangle problems?
2. Can an exterior angle of a triangle be equal to one of its interior angles?
3. What is the sum of all interior angles of a triangle?
4. How can you prove the exterior angle theorem?
5. What would happen if \( \angle PON \) was greater than \( \angle NPO \)?

**Tip:** Remember, the exterior angle theorem is a powerful tool for solving triangle-related problems and finding unknown angles.

Explore how to solve a triangle geometry problem involving exterior angles and interior angles. Understand the application of the Exterior Angle Theorem with detailed step-by-step calculations and solutions. Suitable for high school geometry students.

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