When two parallel lines \( m \) and \( n \) are cut by a transversal, several relationships between the angles formed can be identified based on the properties of parallel lines and transversals.
Given that:
- \( m \parallel n \)
- \( m\angle 5 = 117^\circ \)
We want to find \( m\angle 2 \).
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Identify Angle Relationships:
- Angle \( \angle 5 \) and angle \( \angle 2 \) are corresponding angles. In the case of parallel lines cut by a transversal, corresponding angles are equal. Therefore, if \( \angle 5 = 117^\circ \), then:
\[ m\angle 2 = m\angle 5 = 117^\circ \]
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Conclusion: Since \( m\angle 5 \) and \( m\angle 2 \) are corresponding angles due to the transversal intersecting the parallel lines, we can conclude that:
\[ m\angle 2 = 117^\circ \]
In summary, the value of \( m\angle 2 \) is \( 117^\circ \).