Lenny wrote a paragraph proof of the Perpendicular Bisector Theorem. What mistake did Lenny make in his proof? HK¯¯¯¯¯¯¯¯¯HK¯ is a perpendicular bisector of IJ¯¯¯¯¯¯IJ¯ , and L is the midpoint of IJ¯¯¯¯¯¯IJ¯ . M is a point on the perpendicular bisector, HK¯¯¯¯¯¯¯¯¯HK¯ . By the definition of a perpendicular bisector, I know that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯IM¯≅JM¯ . By the definition of a perpendicular bisector, I also know that ∠MLI∠MLI and ∠MLJ∠MLJ are right angles. ∠MLI≅∠MLJ because of the Right Angle Congruence Theorem. I can also say that ML¯¯¯¯¯¯¯¯¯≅ML¯¯¯¯¯¯¯¯¯ML¯≅ML¯ by the Reflexive Property of Congruence. With this information, I know that △MLI≅△MLJ△MLI≅△MLJ by the SAS Congruence Theorem. Since the triangles are congruent, the CPCTC Theorem allows me to know that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯IL¯≅JL¯ . Knowing that these segments are congruent proves the Perpendicular Bisector Theorem. (1 point) Responses The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯IL¯≅JL¯, not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯IM¯≅JM¯. The definition of a perpendicular bisector tells you that line segment cap i liters is congruent to line segment cap j liters, not that line segment cap i cap m is congruent to line segment cap j cap m. Lenny did not make a mistake. His proof is correct. Lenny did not make a mistake. His proof is correct. The definition of a perpendicular bisector tells you that∠LMI∠LMI and ∠LMJ∠LMJ are right angles, not that ∠MLI∠MLI and ∠MLJ∠MLJ are right angles. The definition of a perpendicular bisector tells you that ∠LMI∠LMI and ∠LMJ∠LMJ are right angles, not that ∠MLI∠MLI and ∠MLJ∠MLJ are right angles.

Question

Lenny wrote a paragraph proof of the Perpendicular Bisector Theorem. What mistake did Lenny make in his proof? HK¯¯¯¯¯¯¯¯¯HK¯ is a perpendicular bisector of IJ¯¯¯¯¯¯IJ¯ , and L is the midpoint of IJ¯¯¯¯¯¯IJ¯ . M is a point on the perpendicular bisector, HK¯¯¯¯¯¯¯¯¯HK¯ . By the definition of a perpendicular bisector, I know that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯IM¯≅JM¯ . By the definition of a perpendicular bisector, I also know that ∠MLI∠MLI and ∠MLJ∠MLJ are right angles. ∠MLI≅∠MLJ because of the Right Angle Congruence Theorem. I can also say that ML¯¯¯¯¯¯¯¯¯≅ML¯¯¯¯¯¯¯¯¯ML¯≅ML¯ by the Reflexive Property of Congruence. With this information, I know that △MLI≅△MLJ△MLI≅△MLJ by the SAS Congruence Theorem. Since the triangles are congruent, the CPCTC Theorem allows me to know that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯IL¯≅JL¯ . Knowing that these segments are congruent proves the Perpendicular Bisector Theorem. (1 point) Responses The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯IL¯≅JL¯, not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯IM¯≅JM¯. The definition of a perpendicular bisector tells you that line segment cap i liters is congruent to line segment cap j liters, not that line segment cap i cap m is congruent to line segment cap j cap m. Lenny did not make a mistake. His proof is correct. Lenny did not make a mistake. His proof is correct. The definition of a perpendicular bisector tells you that∠LMI∠LMI and ∠LMJ∠LMJ are right angles, not that ∠MLI∠MLI and ∠MLJ∠MLJ are right angles. The definition of a perpendicular bisector tells you that ∠LMI∠LMI and ∠LMJ∠LMJ are right angles, not that ∠MLI∠MLI and ∠MLJ∠MLJ are right angles. 
The SAS Congruence Theorem tells you that △MLI≅△JLM△MLI≅△JLM, not that △MLI≅△MLJ△MLI≅△MLJ.

GPT-4o mini · Answer

Lenny's proof contains a couple of mistakes, but the most significant one is related to the angles he identifies as right angles.

The correct observation is that the definition of a perpendicular bisector states that it is a line that is perpendicular to a segment and bisects it. Therefore, the angles created by point M (on the bisector) with the endpoints of segment IJ (I and J) should be referred to as ∠LMI and ∠LMJ, which are the angles formed at point M when you look at segments LM and LI or LJ.

Thus, the incorrect statement in Lenny's proof is when he claims that ∠MLI and ∠MLJ are right angles instead of ∠LMI and ∠LMJ. This is crucial since those angles need to be correctly identified to apply the appropriate congruence argument.

The best choice in the multiple responses you provided is: The definition of a perpendicular bisector tells you that ∠LMI and ∠LMJ are right angles, not that ∠MLI and ∠MLJ are right angles.

This mistake affects the clarity of his definitions and could mislead the structure of the proof related to congruency, ultimately leading to misunderstanding the necessary triangle relationships for demonstrating the Perpendicular Bisector Theorem.