Question

Sylvie has started a proof of the Triangle Angle Sum Theorem. Which answer choice correctly completes her proof? Sylvie's Proof: Given the diagram shown, ∠1≅∠5 ; ∠2≅∠4 because alternate interior angles are congruent when lines are parallel. Then, I know that m∠1=m∠5 ; m∠2=m∠4 because congruent angles have equal measures.
Possible Answers:
m∠1+m∠3+m∠2=180° by the definition of a straight angle. Finally, m∠5+m∠3+m∠4=180° by substitution. m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by the definition of a straight angle. Finally, m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by substitution. m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution. m angle 5 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution. m∠2+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution. m angle 2 plus m angle 3 plus m angle 4 equals 180 degrees by the definition of a straight angle. Finally, m angle 1 plus m angle 3 plus m angle 2 equals 180 degrees by substitution. m∠5+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by the Triangle Angle Sum Theorem.

Answer 1

To complete Sylvie's proof of the Triangle Angle Sum Theorem, we need to focus on the sequence of arguments and ensure coherence in the logic being used.

Sylvie has already established that \( m∠1 = m∠5 \) and \( m∠2 = m∠4 \) due to the congruence of alternate interior angles.

Therefore, the correct answer choice must logically follow from the angles being considered in a triangle and how they sum to 180°.

Among the options given, the most coherent completion of the proof would be:

- "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 sequence utilizes the definitions well and maintains consistency in the logical flow, starting from the straight angle considerations to substitute in the congruent angles that were previously established.