Use the image to answer the question. 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)Responsesm∠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. 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.

Question

Use the image to answer the question.  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)Responsesm∠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. 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.

GPT-4o mini · Answer

To correctly complete Sylvie's proof of the Triangle Angle Sum Theorem, we need to ensure that the statements flow logically based on the properties of angles formed by parallel lines and a transversal.

By Sylvie's reasoning, she established that ∠1 ≅ ∠5 and ∠2 ≅ ∠4 due to alternate interior angles being congruent. Then, she correctly asserted that m∠1 = m∠5 and m∠2 = m∠4 as congruent angles have equal measures.

The next step would be to demonstrate that the sum of the angles in the triangle (formed by angles 1, 2, and the angle at vertex 3) equates to 180°.

The correct response that follows this reasoning 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 completion correctly uses the properties established in the proof and applies the Triangle Angle Sum Theorem appropriately.