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∠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. 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∠2+m∠3+m∠4=180° by the definition of a straight angle. Finally, m∠1+m∠3+m∠2=180° by substitution.

Answer 1

Let's analyze Sylvie's proof to find the correct completion.

Sylvie has established the congruence of angles formed by parallel lines and a transversal:

1. \(\angle 1 \cong \angle 5\)
2. \(\angle 2 \cong \angle 4\)

Then, she states that the measures of these angles are equal:

• \(m\angle 1 = m\angle 5\)
• \(m\angle 2 = m\angle 4\)

Next, she needs to show that the angles summing up to 180 degrees leads to establishing the Triangle Angle Sum Theorem. To do this, she correctly notes that the angles adjacent to angle 3 and forming a straight line add up to 180°.

Considering the responses given, we look for the one that would use the known angle sums appropriately:

Correct Completion:

• "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 choice says that the sum of angles \(m∠5\), \(m∠3\), and \(m∠4\) equals 180° by the definition of a straight angle, which is correct. It then uses substitution to show that \(m∠1 + m∠3 + m∠2\) also equals 180°, leading us to apply the Triangle Angle Sum Theorem appropriately.

Therefore, this is the best completion that aligns with the logic of the proof.