https://file.aiquickdraw.com/mathbot/file/1730299994376q0749ig2.jpg

To find \( m \angle OPN \), let's analyze the triangle given in the image:

1. **Identifying Key Information:**
   - \( \angle PON \) is labeled as \( 122^\circ \).
   - Triangle \( NOP \) has two equal-length sides, indicated by the markings on \( NO \) and \( OP \), suggesting it is an **isosceles triangle** with \( NO = OP \).
   
2. **Isosceles Triangle Property:**
   - In an isosceles triangle, the base angles (angles opposite the equal sides) are equal. Therefore, \( \angle OPN = \angle ONP \).
   
3. **Angle Sum in Triangle \( NOP \):**
   - The sum of the interior angles in any triangle is \( 180^\circ \).
   - Given \( \angle PON = 122^\circ \), we can find \( \angle OPN \) and \( \angle ONP \) as follows:

\[
\angle OPN + \angle ONP + \angle PON = 180^\circ
\]

Substitute \( \angle PON = 122^\circ \):

\[
\angle OPN + \angle ONP + 122^\circ = 180^\circ
\]

Since \( \angle OPN = \angle ONP \), let \( \angle OPN = x \). Then:

\[
x + x + 122^\circ = 180^\circ
\]

\[
2x = 180^\circ - 122^\circ
\]

\[
2x = 58^\circ
\]

\[
x = 29^\circ
\]

4. **Conclusion:**
   - \( m \angle OPN = 29^\circ \).

Would you like more details on any part of this solution, or do you have additional questions?

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Here are 5 related questions to deepen your understanding:

1. How do we know which angles are equal in an isosceles triangle?
2. What is the angle sum property for polygons with more than three sides?
3. How would this problem change if \( \angle PON \) were an acute angle?
4. Could this method apply if all sides were different lengths? Why or why not?
5. What if the angle \( \angle PON \) were an exterior angle instead?

**Tip:** In any isosceles triangle, remember that the angles opposite the equal sides are also equal. This property can simplify solving for unknown angles.

Learn how to find the measure of angle OPN in an isosceles triangle where ∠PON = 122°. This problem explores key geometry concepts such as the isosceles triangle theorem and angle sum property, suitable for Grades 8-10.

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