To complete Sylvie's proof of the triangle angle sum theorem correctly, we need to identify the correct sequence of reasoning that connects the angles based on her previously established equalities due to the parallel lines.
From Sylvie's proof:
- She has established that \( m\angle 1 = m\angle 5 \) and \( m\angle 2 = m\angle 4 \).
- She has also noted that angles around a point (or along a straight line) add up to \( 180^\circ \).
Here's the step-by-step logic:
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Since \( m\angle 5 + m\angle 3 + m\angle 4 = 180^\circ \) is correct due to the definition of a straight angle (the angles on a straight line), we can write: \[ m\angle 5 + m\angle 3 + m\angle 4 = 180^\circ \]
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We can substitute \( m\angle 5 \) and \( m\angle 4 \) with \( m\angle 1 \) and \( m\angle 2 \), respectively: \[ m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ \]
Now we can look at the answer choices:
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Option A:
- \( m\angle 5 + m\angle 3 + m\angle 4 = 180^\circ \) (correct)
- \( m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ \) (correct by substitution from congruent angles)
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Option B:
- It states the angles incorrectly in the final statement by switching the order. This is not the direct and proper transition from the first step.
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Option C:
- It has the same equations but doesn't apply substitution correctly in the order.
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Option D:
- It incorrectly uses \( m\angle 2 + m\angle 3 + m\angle 4 \) and leads incorrectly to the triangle angle sum theorem without sufficient logical connection.
The best answer that completes Sylvie's proof is: A. \( m\angle 5 + m\angle 3 + m\angle 4 = 180^\circ \) by the definition of a straight angle. Finally, \( m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ \) by the triangle angle sum theorem.
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