To complete Sylvie's proof of the Triangle Angle Sum Theorem correctly, we first note that she has established that \( m∠1 = m∠5 \) and \( m∠2 = m∠4 \). Then, using the definition of a straight angle, we understand that angles \( ∠5 \), \( ∠3 \), and \( ∠4 \) form a linear pair, and therefore:
\[ m∠5 + m∠3 + m∠4 = 180° \]
From there, she aims to eventually conclude about the angles \( ∠1 \), \( ∠2 \), and \( ∠3 \) that form a triangle, showing that:
\[ m∠1 + m∠3 + m∠2 = 180° \]
Now we can match the completion of her proof to the correct response. The most logical choice complete Sylvie’s proof, after establishing the angle measures and concluding with the triangle angle sum, is:
m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.
This option correctly follows from the previous conclusions and reaches the needed conclusion of the Triangle Angle Sum Theorem.
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