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.

A. The SAS Congruence Theorem tells you that △MLI≅△JLM , not that △MLI≅△MLJ.
B. The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯ , not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯ .
C. The definition of a perpendicular bisector tells you that∠LMI and ∠LMJ are right angles, not that ∠MLI and ∠MLJ are right angles.
D. Lenny did not make a mistake. His proof is correct.

Answer 1

The main mistake in Lenny's proof is option C.

Lenny states that ∠MLI and ∠MLJ are right angles due to HK being the perpendicular bisector, which is correct. However, he should have explicitly stated that ∠LMI and ∠LMJ are the right angles formed with the segments rather than incorrectly labeling them as ∠MLI and ∠MLJ. The proper naming of the angles is crucial for clarity in the proof, and the confusion may affect the logical flow of the proof regarding the triangles' congruence.

So the accurate answer regarding the mistake that Lenny made in his proof is:

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