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.

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The mistake in Lenny's proof lies in the reasoning regarding the angles and which triangles are being considered. Specifically, he incorrectly identifies the angles involved in the right angle definition of a perpendicular bisector.

The correct angles in the context should be ∠LMI and ∠LMJ, not ∠MLI and ∠MLJ. Thus, the proper angle notation must be applied to emphasize the angles at point L with respect to the segments from the midpoint to the endpoints of segment IJ.

Therefore, the correct 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.

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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.

1 answer

The mistake in Lenny's proof lies in the reasoning regarding the angles and which triangles are being considered. Specifically, he incorrectly identifies the angles involved in the right angle definition of a perpendicular bisector.

The correct angles in the context should be ∠LMI and ∠LMJ, not ∠MLI and ∠MLJ. Thus, the proper angle notation must be applied to emphasize the angles at point L with respect to the segments from the midpoint to the endpoints of segment IJ.

Therefore, the correct 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.

GPT-4o mini · Answer

The mistake in Lenny's proof lies in the reasoning regarding the angles and which triangles are being considered. Specifically, he incorrectly identifies the angles involved in the right angle definition of a perpendicular bisector.

The correct angles in the context should be ∠LMI and ∠LMJ, not ∠MLI and ∠MLJ. Thus, the proper angle notation must be applied to emphasize the angles at point L with respect to the segments from the midpoint to the endpoints of segment IJ.

Therefore, the correct 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.