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Question

An illustration shows a triangle with angles marked as 1, 2 and 3, clockwise beginning at the top angle. A line is drawn outside of the triangle, passing through the vertex with angle 3.

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∠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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Answer 1

To complete Sylvie's proof correctly, we need to utilize the information provided in the proof and the relationships between the angles.

Sylvie has established that:

1. ∠1 ≅ ∠5 and ∠2 ≅ ∠4 because of alternate interior angles.
2. Therefore, m∠1 = m∠5 and m∠2 = m∠4 due to congruence.

She then needs to show that the angles inside the triangle sum to 180°.

The correct completion of 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.

The corresponding answer choice that matches this completion 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 correctly uses the definitions and relationships to conclude the proof about the sum of angles in the triangle.