Statement 1: angle K = angle N
Reason 1: Given
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Statement 2: segment LM bisects angle KLN
Reason 2: Given
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Statement 3: Angle KLM = angle NLM
Reason 3: Definition of angle bisector
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Statement 4: LM = LM
Reason 4: Reflexive property of congruence
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Statement 5: Triangle KLM = Triangle NLM
Reason 5: AAS congruence theorem
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Statement 6: LK = LN
Reason 6: CPCTC
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note: CPCTC stands for "corresponding parts of congruent triangles are congruent". Its like saying "those two houses are identical, so the front doors must be the same". In this analogy, the houses represent the triangles while the front doors represent the side lengths. If we know both triangles are the same, then the corresponding matching pieces must be the same as well (otherwise the triangles would be different).
see the attached image below for how the proof would look in a table format.