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

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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1 answer

In Lenny's proof of the Perpendicular Bisector Theorem, the mistake he made can be identified in his interpretation of the angles created by the perpendicular bisector. Specifically, he states that "∠MLI and ∠MLJ are right angles," while it is actually the angles adjacent to point L (where the perpendicular bisector intersects IJ) that should be referenced.

The correct notation should be:

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

Therefore, the correct response indicating the mistake is: "The definition of a perpendicular bisector tells you that ∠LMI and ∠LMJ are right angles, not that ∠MLI and ∠MLJ are right angles."

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