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To find the measure of angle \( m \angle MON \), let’s analyze the triangle \( \triangle MON \) based on the information provided:

1. The triangle \( \triangle MON \) is an **isosceles triangle**, as indicated by the markings on sides \( ON \) and \( NM \), which show they are of equal length.
2. Since \( \triangle MON \) is isosceles, the angles opposite these equal sides, \( \angle MON \) and \( \angle MNO \), must also be equal.
3. The measure of \( \angle NMO \) is given as \( 57^\circ \).

### Solution Steps
1. Let \( m \angle MON = m \angle MNO = x \).
2. The sum of the interior angles of a triangle is always \( 180^\circ \), so we have:
   \[
   x + x + 57^\circ = 180^\circ
   \]
   \[
   2x + 57^\circ = 180^\circ
   \]
3. Subtract \( 57^\circ \) from both sides:
   \[
   2x = 123^\circ
   \]
4. Divide by 2 to find \( x \):
   \[
   x = 61.5^\circ
   \]

### Conclusion
The measure of \( m \angle MON \) is \( 61.5^\circ \).

Would you like more details on any of these steps?

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Here are some related questions to explore further:

1. How would the solution change if \( \angle NMO \) were different?
2. How can we verify that a triangle is isosceles by using its angle measures?
3. What are the properties of isosceles triangles that help simplify angle calculations?
4. How does the triangle sum theorem apply to other polygon types?
5. How can we find the exterior angles of an isosceles triangle?

**Tip:** Always look for symmetry in triangles to simplify calculations, especially in isosceles and equilateral triangles.

Find m∠MON in the given isosceles triangle where m∠NMO = 57°.

Learn how to find the measure of angle MON in an isosceles triangle with a given angle of 57°. This problem covers essential geometry concepts, including isosceles triangle properties and the triangle angle sum theorem. Ideal for Grades 7-9.

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