Question

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 SAS Congruence Theorem tells you that △MLI≅△JLM , not that △MLI≅△MLJ . The SAS Congruence Theorem tells you that △MLI≅△JLM , not that △MLI≅△MLJ . The definition of a perpendicular bisector tells you that∠LMI and ∠LMJ are right angles, not that ∠MLI and ∠MLJ are right angles.

Answer 1

The mistake in Lenny's proof lies in how he identifies the angles. Lenny states that ∠MLI and ∠MLJ are right angles. However, because L is the midpoint of IJ, it should be the angles ∠LMI and ∠LMJ that are the right angles, as these are the angles formed by the perpendicular bisector line HK and the segments LI and LJ.

So, the correct response is:

"The definition of a perpendicular bisector tells you that ∠LMI and ∠LMJ are right angles, not that ∠MLI and ∠MLJ are right angles."