Proofs About Angles Quick Check

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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
a b c or d which one is truly correct and why or why not
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

Let's analyze Lenny's proof step by step to identify the mistake in his reasoning regarding the Perpendicular Bisector Theorem.

From the given proof:

1. Lenny correctly establishes that \( HK \) is a perpendicular bisector of \( IJ \) and identifies \( L \) as the midpoint of \( IJ \).
2. He states that by the definition of a perpendicular bisector, \( IM \cong JM \). This is correct because any point \( M \) on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.
3. He states that \( \angle MLI \) and \( \angle MLJ \) are right angles. This is also correct because the definition of a perpendicular bisector implies that it forms right angles with the segment it bisects.
4. Then he claims \( \angle MLI \cong \angle MLJ \) through the Right Angle Congruence Theorem. This is valid as both angles are right angles.
5. He states that \( ML \cong ML \) by the Reflexive Property of Congruence, which is also correct.
6. Next, Lenny claims that \( \triangle MLI \cong \triangle MLJ \) by the SAS Congruence Theorem. Here is where Lenny makes his primary mistake: he incorrectly identifies the two triangles he is comparing. The corresponding triangles should be \( \triangle MLI \) and \( \triangle MLJ \), but he mixes up the labels and uses \( \triangle MLI \) and \( \triangle JLM \) instead.

The more relevant error related to congruence used in the SAS Congruence Theorem should indeed be recognizing the structure of the triangles correctly.

Given these points, let’s evaluate the responses:

The correct response would be:

• "The SAS Congruence Theorem tells you that \( \triangle MLI \cong \triangle JLM \), not that \( \triangle MLI \cong \triangle MLJ \)."

Thus, the correct choice is: The SAS Congruence Theorem tells you that \( \triangle MLI \cong \triangle JLM \), not that \( \triangle MLI \cong \triangle MLJ \). This choice reflects Lenny's fundamental misunderstanding of which triangles are actually being compared for congruency.

If you have further questions, feel free to ask!