Lenny wrote a paragraph proof of the Perpendicular Bisector Theorem. What mistake did Lenny make in his proof?

Question

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∠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 SAS Congruence Theorem tells you that △MLI≅△JLM, not that △MLI≅△MLJ.

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.

Answer 1

Lenny made the following mistake in his proof:

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

In the context of the perpendicular bisector theorem, when a line is described as a perpendicular bisector, it means that it intersects the segment at its midpoint and forms right angles with it. Therefore, the angles at point L should be referenced relative to segment IJ, specifically angles ∠LMI and ∠LMJ, rather than ∠MLI and ∠MLJ, which are incorrectly labeled.