To fill in the blanks in Aaliyah's explanation, we can analyze the statements made and find suitable options for each one.
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The first blank refers to the reason Aaliyah used to conclude that triangle KLM is congruent to triangle MNK, which is established based on the information that LM ≅ NK and KM ≅ MK. Since two sides of both triangles are congruent along with the included angle (∠LMK ≅ ∠NKM), we can use the SAS (Side-Angle-Side) congruence theorem.
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The second blank will be the property or theorem used to justify that KM ≅ MK. Given that these are two sides of a parallelogram and the opposite sides of a parallelogram are congruent, we can justify this using the property of the parallelogram rather than one of the provided options. But here, we assume that we need to fill with one of the options. The closest fitting might be the Reflexive Property of Congruence, although KM and MK refer to the same segment.
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In the final blank, when concluding that LK ≅ NM follows from the congruence of triangles KLM and MNK, we refer to CPCTC (Corresponding Parts of Congruent Triangles are Congruent), as it states that if two triangles are congruent, then their corresponding parts are also congruent.
With these analyses, the filled explanation would read:
It is given that LM ≅ NK and Option #3. I also know that KM ≅ MK by Option #1. This means that △KLM ≅ △MNK by the SAS Congruence Theorem. Therefore, LK ≅ NM by Option #2.
So, the answers to fill in are:
- First blank (Option #) should be Option #3 (the SAS Congruence Theorem).
- Second blank (Option #) should be Option #1 (the Reflexive Property of Congruence).
- Third blank (Option #) should be Option #2 (the CPCTC Theorem).