Question

Question
Use the image to answer the 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 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.
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

To correctly complete Sylvie's proof of the Triangle Angle Sum Theorem, we need to follow the steps and logic presented.

Based on Sylvie's proof, we have already established:

1. The measures of angles ∠1 and ∠5 are equal (m∠1 = m∠5) due to alternate interior angles being congruent when lines are parallel.
2. The measures of angles ∠2 and ∠4 are equal (m∠2 = m∠4) for the same reason.

Now, since ∠5 and ∠3 and ∠4 are on a straight line (assuming line passes through angle ∠3), we can apply the definition of a straight angle:

m∠5 + m∠3 + m∠4 = 180°.

Then, using the equalities we established:

m∠5 + m∠3 + m∠4 = 180°.

Now, substituting m∠5 with m∠1 and m∠4 with m∠2 gives:

m∠1 + m∠3 + m∠2 = 180°.

Thus, the correct option to complete Sylvie's 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 the Triangle Angle Sum Theorem.**

This shows that both expressions are equal to 180°, confirming the Triangle Angle Sum Theorem.

So the correct answer choice is the one that states:
"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."