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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.

1 answer

To correctly complete Sylvie's proof of the Triangle Angle Sum Theorem, we need a response that accurately follows from her previous statements and connects the angles in a way that aligns with the proofs involving congruent angles and straight angles.

Based on her reasoning, she established that:

  1. ∠1≅∠5 and ∠2≅∠4 due to the properties of parallel lines.
  2. She equated the measures of these angles due to their congruence.

From there, the logical next step in her proof should involve utilizing the fact that the angles of a triangle add up 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.

This option correctly states that the sum of angles adjacent to a straight angle is 180 degrees and aligns logically with her earlier conclusions about the angles. It appropriately concludes the proof by showing that the angles in a triangle satisfy the Triangle Angle Sum Theorem.