Sylvie has started a proof of the Triangle Angle Sum Theorem. Which answer choice correctly completes her proof?

Question

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∠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∠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∠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, we need to logically follow the reasoning regarding the angles and their relationships. Given that ∠1 and ∠5 are congruent, along with ∠2 and ∠4, the proof will reference the relationships between these angles and their measures.

The correct choice should begin with stating that the sum of angles around point where ∠3 is located (on a straight line) equals 180°. This means we can use one of the angles formed by the transversal and the parallel lines to arrive at the necessary conclusions.

The right conclusion after stating that the measures of angles add up to 180° would include substitution based on the congruence established earlier.

The best 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 is correct because it follows from the definition of a straight angle, and then it substitutes the congruent angles into the equation to reach the conclusion needed for the Triangle Angle Sum Theorem.