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Use the image to answer the question.

A parallelogram K L M N. Vertices K and L are on the top left and right, respectively. Vertices L and M are on the bottom left and right, respectively.

Aaliyah has written an explanation for why LK¯¯¯¯¯¯¯¯≅NM¯¯¯¯¯¯¯¯¯¯
. Choose the correct option to fill in each blank.

Option #1: the Reflexive Property of Congruence

Option #2: the CPCTC Theorem

Option #3: the SAS Congruence Theorem

Option #4: ∠LMK≅∠NKM


(2 points)
It is given that LM¯¯¯¯¯¯¯¯¯≅NK¯¯¯¯¯¯¯¯¯
and Option #
. I also know that KM¯¯¯¯¯¯¯¯¯¯≅MK¯¯¯¯¯¯¯¯¯¯
by Option #
. This means that △KLM≅△MNK
by the SAS Congruence Theorem. Therefore, LK¯¯¯¯¯¯¯¯≅NM¯¯¯¯¯¯¯¯¯¯
by Option #
.

1 answer

To fill in the blanks correctly based on the information you provided about the parallelogram KLMN, we can analyze what is given and what needs to be concluded.

  1. You have the statement that \( LM \cong NK \).
  2. To prove triangles \( KLM \) and \( MNK \) are congruent using the SAS (Side-Angle-Side) Congruence Theorem, you first need to identify the sides and the angle.

Given that triangles \( KLM \) and \( MNK \) share a side (\( KM \) and \( MK \) are congruent by reflexivity of segments), we can fill in the options as follows:

So the filled-in passage becomes:

"It is given that \( LM \cong NK \) and Option #1: the Reflexive Property of Congruence. I also know that \( KM \cong MK \) by Option #1: the Reflexive Property of Congruence. This means that \( \triangle KLM \cong \triangle MNK \) by the Option #3: the SAS Congruence Theorem. Therefore, \( LK \cong NM \) by Option #2: the CPCTC Theorem."

This way, you can fill the options as follows:

  • For the first blank (after "and"): Option #1: the Reflexive Property of Congruence
  • For the second blank (after "by"): Option #1: the Reflexive Property of Congruence
  • For the third blank (before "by"): Option #3: the SAS Congruence Theorem
  • For the last blank (after "by"): Option #2: the CPCTC Theorem