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 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.
  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 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.
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GPT-4o mini · Answer

To complete Sylvie's proof of the Triangle Angle Sum Theorem, we need to recognize the relationships between the angles she has established.

Given:

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

From her statements, we also know:

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

Sylvie should then establish that the measures of angles around a straight line sum to \( 180^\circ \).

The most logical conclusion from the established congruences would be:

\( m\angle 5 + m\angle 3 + m\angle 4 = 180^\circ \) by the definition of a straight angle.

Since she can then substitute the congruent angles:

\( m\angle 1 + m\angle 3 + m\angle 2 = 180^\circ \) by substitution.

Thus, the correct option that completes her 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.

So the best response is: 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.