Question

1. fill in the missing statements and reasons to complete the following proof.

given: l is the midpoint of \\(\overline{jn}\\), \\(\angle k \cong \angle m\\)
prove: \\(\overline{jk} \cong \overline{nm}\\)
image of a geometric figure with points j, k, l, n, m and a proof table with missing parts

Explanation:

Step1: Identify Given Information

Statement 1: \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \) (Given)

Step2: State Midpoint Definition

Statement 3: \( JL = LN \) (Definition of Midpoint, since \( L \) is midpoint of \( \overline{JN} \))

Step3: Identify Vertical Angles

Statement 4: \( \angle KLJ \cong \angle MLN \) (Vertical Angle Theorem)

Step4: Prove Triangle Congruence

Statement 5: \( \triangle KLJ \cong \triangle MLN \) (AAS: \( \angle K \cong \angle M \), \( \angle KLJ \cong \angle MLN \), \( JL = LN \))

Step5: Corresponding Parts of Congruent Triangles

Statement 7: \( \overline{JK} \cong \overline{NM} \) (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Answer:

1. \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \)
2. (Reason: Given for \( \angle K \cong \angle M \))
3. \( JL = LN \)
4. \( \angle KLJ \cong \angle MLN \)
5. \( \triangle KLJ \cong \triangle MLN \) (AAS)
6. (Reason: AAS Congruence)
7. \( \overline{JK} \cong \overline{NM} \) (CPCTC)
8. (Reason: CPCTC)

(Note: The exact numbering and labels may vary based on the proof structure, but the key steps involve AAS congruence and CPCTC to show the sides are congruent.)