Question

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
which answer is correct and why by explaining and solving in paragraph
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.
The definition of a perpendicular bisector tells you that ∠LMI
and ∠LMJ
are right angles, not that ∠MLI
and ∠MLJ
are right angles.

Lenny did not make a mistake. His proof is correct.
Lenny did not make a mistake. His proof is correct.
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Answer 1

Lenny made a mistake in his proof regarding the identification of right angles in the triangle. Specifically, the correct conclusion regarding the right angles should involve the angles ∠LMI and ∠LMJ, rather than ∠MLI and ∠MLJ.

The correct statements about the perpendicular bisector of segment IJ relate to point L as the midpoint, creating two segments, IL and JL, that are equal. When Lenny claims that ∠MLI and ∠MLJ are right angles based on the definition of the perpendicular bisector, this is incorrect. The perpendicular bisector implies that angles ∠LMI and ∠LMJ are the right angles, as HK is the line that bisects IJ at L and is perpendicular to it. This distinction is critical,

Thus, the right 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.

By misidentifying the angles, Lenny's proof lacks clarity in establishing the congruence of the triangles correctly, which is necessary for his conclusion to be valid. Therefore, this mistake weakens the logic of the proof.