lenny wrote a paragraph proof of the perpendicular bisector theorem. what mistake did lenny make in his proof?
$ \overline { h k } $ is a perpendicular bisector of $ \overline { i j } $, and $ l $ is the midpoint of $ \overline { i j } $. $ m $ is a point on the perpendicular bisector, $ \overline { h k } $. by the definition of a perpendicular bisector, i know that $ \overline { i m } \cong \overline { j m } $. by the definition of a perpendicular bisector, i also know that $ \angle m l i $ and $ \angle m l j $ are right angles. $ \angle m l i \cong \angle m l j $ because of the right angle congruence theorem. i can also say that $ \overline { m l } \cong \overline { m l } $ by the reflexive property of congruence. with this information, i know that $ \triangle m l i \cong \triangle m l j $ by the sas congruence theorem. since the triangles are congruent, the cpctc theorem allows me to know that $ \overline { i l } \cong \overline { j l } $. knowing that these segments are congruent proves the perpendicular bisector theorem.
(1 point)
the sas congruence theorem tells you that $ \triangle m l i \cong \triangle j l m $, not that $ \triangle m l i \cong \triangle m l j $
lenny did not make a mistake. his proof is correct.
the definition of a perpendicular bisector tells you that $ \overline { i l } \cong \overline { j l } $, not that $ \overline { i m } \cong \overline { j m } $
the definition of a perpendicular bisector tells you that $ \angle l m i $ and $ \angle l m j $ are right angles, not that $ \angle m l i $ and $ \angle m l j $ are right angles

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Accepted Answer

# Answer:
The definition of a perpendicular bisector tells you that $\overline{IL}\cong\overline{JL}$, not that $\overline{IM}\cong\overline{JM}$

# Explanation:
## Step1: Recall the definition of perpendicular bisector
The perpendicular bisector of a segment is a line that is perpendicular to the segment and passes through its mid - point. By the definition of a perpendicular bisector, if $HK$ is the perpendicular bisector of $IJ$ and $L$ is the mid - point of $IJ$, then $IL = JL$ (i.e., $\overline{IL}\cong\overline{JL}$).
## Step2: Analyze the incorrect statement
The statement “By the definition of a perpendicular bisector, I know that $\overline{IM}\cong\overline{JM}$” is wrong. The property $\overline{IM}\cong\overline{JM}$ (where $M$ is a point on the perpendicular bisector $HK$) is the conclusion of the Perpendicular Bisector Theorem (a point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment), not the definition. The definition only gives the congruence of the two sub - segments ($\overline{IL}$ and $\overline{JL}$) of the bisected segment $IJ$ by its mid - point $L$.

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Explanation:

Step1: Recall the definition of perpendicular bisector

Step2: Analyze the incorrect statement

Answer: