Lenny made a mistake in his proof regarding the orientation of angles in the statement about the definition of a perpendicular bisector. Specifically, he incorrectly stated that ∠MLI and ∠MLJ are right angles instead of ∠LMI and ∠LMJ. The definition of a perpendicular bisector indicates that it forms right angles with the segment it bisects at its midpoint. In this case, since L is the midpoint of segment IJ, the correct angles to refer to should be ∠LMI and ∠LMJ, which are indeed the angles formed at point L by the perpendicular bisector HK with segments IL and JL. Therefore, the error lies in the labeling of the angles, which leads to confusion regarding the relationship between the segments and angles in the triangles formed. This misrepresentation of the angles impacts the correctness of subsequent statements regarding the congruence of triangles. As a result, statements that rely on this definition of the angles will also be flawed. Thus, Lenny's proof cannot be considered correct as it stands due to this pivotal labeling mistake. To ensure clarity and accuracy, it is crucial to correctly identify the angles created by the perpendicular bisector at the midpoint of the segment.
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 in a 10 sentence 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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