It seems you’re working through a proof involving parallel lines and angles. Let's clarify and complete the reasoning step-by-step based on your statements and the properties of angles formed by parallel lines.
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Given: You have two parallel lines cut by a transversal, and you identified several angles.
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Angles 1 and 5: You've stated that \( \angle 1 \cong \angle 5 \). This would be true if \( \angle 1 \) and \( \angle 5 \) are corresponding angles (which they can be if they are created by a transversal cutting through two parallel lines). Therefore, \( m\angle 1 = m\angle 5 \).
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Angles 2 and 4: You've noted that \( \angle 2 \cong \angle 4 \) due to the Alternate Interior Angles Theorem, which is correct when two lines are parallel. Hence, \( m\angle 2 = m\angle 4 \).
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Straight Angle Property: You mentioned that \( m\angle 1 + m\angle 3 + m\angle 4 = 180^\circ \). This is a reflection of the fact that angles on a straight line sum to \( 180^\circ \).
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Substitution of Angle Measures: Based on the equal measures established earlier, you can substitute for \( m\angle 4 \) in your above equation using \( m\angle 2 \) (since \( m\angle 2 = m\angle 4 \)):
- Since \( m\angle 4 = m\angle 2 \), rewrite the equation as: \[ m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ \]
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Using the Equality of Angles: Since you've established that \( m\angle 1 = m\angle 5 \) (from step 2), you could also write: \[ m\angle 5 + m\angle 3 + m\angle 2 = 180^\circ \]
By rearranging any of these equations, you can draw conclusions about relationships between the angles based on their measures being equal and the properties of the angles formed by parallel lines and transversals.
This proof highlights the consistency of the properties of parallel lines and angles:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Angles on a straight line sum to \( 180^\circ \).
If you need more help or a specific step clarified, feel free to ask!