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. (1 point) Responses 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. 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 the Triangle Angle Sum Theorem. m∠1+m∠3+m∠2=180° by the definition of a straight angle. Finally, m∠5+m∠3+m∠4=180° by substitution.

Answer 1

To complete Sylvie's proof of the Triangle Angle Sum Theorem, the logical steps need to follow from the initial congruences of angles and the definition of a straight angle. The appropriate choice here should correctly conclude the proof based on the properties of angles when summing them up.

Given that Sylvie has established that:

1. \(\angle 1 \cong \angle 5\) and \(\angle 2 \cong \angle 4\),
2. That measures of congruent angles are equal,
3. Angles on a straight line sum to 180°,

The correct answer should emphasize that the angles formed by a straight angle equal 180° and relate that back to the triangle's angles.

Therefore, the correct completion of the proof 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 response correctly uses the definition of a straight angle and solidly concludes with the Triangle Angle Sum Theorem.