To solve this problem, we need to use the properties of parallel lines and the angles formed by a transversal.
Given that JL and MO are parallel lines and \( mLKI = 116^\circ \), we can observe the configuration of angles formed when a transversal crosses these parallel lines.
Here's how we can deduce \( mONP \):
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Identify angles: Since JL and MO are parallel and assuming that the transversal intersects these lines, then angle LKI (which is inside the transversal lines) will have a corresponding angle formed on the opposite side of the transversal with angle ONP.
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Calculate \( mONP \): By the properties of corresponding angles, \( mLKI + mONP = 180^\circ \) (since they are supplementary). Therefore, we can write:
\[ mONP = 180^\circ - mLKI \]
Substituting the value of \( mLKI\):
\[ mONP = 180^\circ - 116^\circ = 64^\circ \]
Thus, \( mONP = 64^\circ \).