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∠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∠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∠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

To complete Sylvie's proof of the Triangle Angle Sum Theorem, we need to establish relations based on the definitions and properties stated. The correct completion must show that:

1. The angles in a triangle sum to 180°.
2. Appropriate substitutions or references to previously established congruences must be made.

The reasoning follows that if we have a transversal cutting two parallel lines, we can use alternate interior angles and the fact that angles on a straight line sum to 180°.

Considering the details provided in Sylvie's proof, the correct completion would involve deriving the sums of the angles of the triangle from the straight angle formed by those measures.

The correct answer choice 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 choice correctly states that the sum of the angles including the angle at the vertex (∠3) and the corresponding alternate interior angles equals 180°, and it uses substitution to relate the measures of angles in the triangle to those angles formed by the parallel lines and a transversal.