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