Question

aaliyah has written an explanation for why ( overline{lk} cong overline{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: ( angle lmk cong angle nkm )
(2 points)
it is given that ( overline{lm} cong overline{nk} ) and option #. i also know that ( overline{km} cong overline{mk} ) by option #.
this means that ( \triangle klm cong \triangle mnk ) by the sas congruence theorem. therefore,
( overline{lk} cong overline{nm} ) by option #.

Explanation:

Step1: Identify the angle for SAS

For the SAS (Side - Angle - Side) Congruence Theorem, we need two sides and the included angle. We are given \(\overline{LM}\cong\overline{NK}\) and \(\overline{KM}\cong\overline{MK}\). The included angle for the two triangles \(\triangle KLM\) and \(\triangle MNK\) is \(\angle LMK\cong\angle NKM\) (Option #4).

Step2: Use the Reflexive Property

The Reflexive Property of Congruence states that a segment is congruent to itself. So, \(\overline{KM}\cong\overline{MK}\) by the Reflexive Property of Congruence (Option #1).

Step3: Use CPCTC

CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is used to show that corresponding parts of congruent triangles are congruent. Since \(\triangle KLM\cong\triangle MNK\) by SAS, then \(\overline{LK}\cong\overline{NM}\) by CPCTC (Option #2).

Answer:

First blank: 4, Second blank: 1, Third blank: 2