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 diagram with triangles and a line segment jn, with points k, l, m, n

1. blank → 3. blank

given definition of midpoint
\\(\angle k \cong \angle m\\)

1. blank
2. blank → 5. blank
3. blank → 7. blank
4. blank

list of terms: cpctc, reflexive property, transitive property, substitution, definition of midpoint, vertical angles theorem, linear pair, definition of segment bisector
blank 1 to blank 8 with checkboxes for each term

Explanation:

Step1: Identify Given Information

The first blank (Blank 1) is labeled "Given", so we use the given information: \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \) (but the first given is \( L \) is midpoint of \( \overline{JN} \), so Blank 1: \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \) (wait, no, the first "Given" box: the problem says "Given: \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \)", so Blank 1 statement is " \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \)" (but maybe split? Wait, the diagram has Blank 1 as "Given" with the first given. Wait, the first given is \( L \) is midpoint of \( \overline{JN} \), and \( \angle K \cong \angle M \) is another given. Wait, the flow:

1. **Blank 1 (Statement, Given)**: \( L \) is the midpoint of \( \overline{JN} \) (and \( \angle K \cong \angle M \) is another given, maybe Blank 2? Wait, the diagram:

- Blank 1: Statement, Given: so " \( L \) is the midpoint of \( \overline{JN} \)" (since the other given is \( \angle K \cong \angle M \) in the box below)
- Blank 2: Statement, maybe \( \angle KLJ \cong \angle MLN \) (Vertical Angles), reason: Vertical Angle Theorem (Blank 4)
- Blank 3 (Statement, Definition of Midpoint): \( JL = LN \) (since midpoint divides into two equal segments)
- Blank 4 (Statement, Vertical Angle Theorem): \( \angle KLJ \cong \angle MLN \)
- Then, to prove triangles congruent: \( \triangle KLJ \cong \triangle MLN \) (AAS: \( \angle K \cong \angle M \), \( \angle KLJ \cong \angle MLN \), \( JL = LN \))
- Then Blank 7: \( \overline{JK} \cong \overline{NM} \) (CPCTC)
- Blank 6: Reason for triangle congruence, maybe AAS (Angle - Angle - Side)
- Blank 5: Statement: \( \triangle KLJ \cong \triangle MLN \)
- Blank 8: Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Wait, let's re - structure:

Step1: Blank 1 (Statement, Given)

The first given is \( L \) is the midpoint of \( \overline{JN} \), so Blank 1: \( L \) is the midpoint of \( \overline{JN} \) (and \( \angle K \cong \angle M \) is another given, but the first "Given" box is for the midpoint? Wait, the problem says "Given: \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \)". So Blank 1 (statement) is " \( L \) is the midpoint of \( \overline{JN} \)", and the other given is \( \angle K \cong \angle M \) in the box below (Blank 2? No, the box with \( \angle K \cong \angle M \) is already a statement with no label, maybe Blank 2 is the reason? No, the blanks are numbered 1 - 8. Let's list the blanks:

1. Blank 1: Statement, Given: \( L \) is the midpoint of \( \overline{JN} \)
2. Blank 2: Maybe the reason for \( \angle K \cong \angle M \), but it's given, so no. Wait, the "Vertical Angle Theorem" is for Blank 4 (statement: \( \angle KLJ \cong \angle MLN \))
3. Blank 3: Statement, Definition of Midpoint: \( JL = LN \) (since midpoint means \( JL = LN \))
4. Blank 4: Statement, Vertical Angle Theorem: \( \angle KLJ \cong \angle MLN \)
5. Blank 5: Statement: \( \triangle KLJ \cong \triangle MLN \) (by AAS: \( \angle K \cong \angle M \), \( \angle KLJ \cong \angle MLN \), \( JL = LN \))
6. Blank 6: Reason: AAS (Angle - Angle - Side) Congruence Criterion
7. Blank 7: Statement: \( \overline{JK} \cong \overline{NM} \) (CPCTC)
8. Blank 8: Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Now, let's check the options (the bottom has options: CPCTC, Reflexive Property, Transitive Property, Substitution, Definition of midpoint, Vertical Angles Theorem, Linear Pair, Definition of Segment Bisector)

- Blank 3 (Definition of Midpoint): Statement is \( JL = LN \) (so Blank 3: \( JL = LN \))
- Blank 4 (Vertical Angle Theorem): Statement is \( \angle KLJ \cong \angle MLN \)
- Blank 5 (Statement): \( \triangle KLJ \cong \triangle MLN \)
- Blank 6 (Reason): AAS (but the options: wait, the options are CPCTC, Reflexive, Transitive, Substitution, Definition of midpoint, Vertical Angles Theorem, Linear Pair, Definition of Segment Bisector. Wait, maybe the congruence is AAS, but the options don't have AAS? Wait, maybe I made a mistake. Wait, the given: \( \angle K \cong \angle M \) (angle), \( JL = LN \) (side), and \( \angle KLJ \cong \angle MLN \) (angle) → AAS. But the options for reasons: let's see the blanks 6 and 8.

Blank 8: Reason for \( \overline{JK} \cong \overline{NM} \) is CPCTC (so Blank 8: CPCTC)

Blank 6: Reason for \( \triangle KLJ \cong \triangle MLN \) → AAS, but the options: maybe the options are mis - labeled? Wait, the bottom has "Blank 6" with a circle on "Reflexive Property"? No, maybe the correct steps:

1. **Blank 1 (Statement, Given)**: \( L \) is the midpoint of \( \overline{JN} \), \( \angle K \cong \angle M \) (but split: first given is \( L \) midpoint, second is \( \angle K \cong \angle M \))
2. **Blank 2 (Statement)**: \( \angle KLJ \cong \angle MLN \) (Vertical Angles)
3. **Blank 3 (Statement, Definition of Midpoint)**: \( JL = LN \)
4. **Blank 4 (Reason)**: Vertical Angle Theorem (for \( \angle KLJ \cong \angle MLN \))
5. **Blank 5 (Statement)**: \( \triangle KLJ \cong \triangle MLN \) (by AAS: \( \angle K \cong \angle M \), \( \angle KLJ \cong \angle MLN \), \( JL = LN \))
6. **Blank 6 (Reason)**: AAS (but options: maybe the options have "AAS" but in the given options, maybe it's a typo. Alternatively, maybe the congruence is ASA? No, \( \angle K \cong \angle M \), \( JL = LN \), \( \angle KLJ \cong \angle MLN \) is AAS.

7. **Blank 7 (Statement)**: \( \overline{JK} \cong \overline{NM} \)
8. **Blank 8 (Reason)**: CPCTC

But since the problem is about triangle congruence proof, the subfield is **Geometry** (Mathematics).

Now, filling the blanks:

- Blank 1: \( L \) is the midpoint of \( \overline{JN} \) (Given)
- Blank 2: \( \angle KLJ \cong \angle MLN \) (Statement, to be proven by Vertical Angle Theorem)
- Blank 3: \( JL = LN \) (Statement, Definition of Midpoint)
- Blank 4: \( \angle KLJ \cong \angle MLN \) (Statement, Vertical Angle Theorem)
- Blank 5: \( \triangle KLJ \cong \triangle MLN \) (Statement, by AAS)
- Blank 6: AAS (Angle - Angle - Side) Congruence (but in the options, maybe "ASA" is not there, but the closest. Wait, the options given: CPCTC, Reflexive Property, Transitive Property, Substitution, Definition of midpoint, Vertical Angles Theorem, Linear Pair, Definition of Segment Bisector. Maybe the problem has a different approach. Wait, the goal is to prove \( \overline{JK} \cong \overline{NM} \), so triangles \( \triangle JKL \) and \( \triangle NML \)? Wait, maybe the triangles are \( \triangle JKL \) and \( \triangle NML \), with \( JL = LN \) (midpoint), \( \angle K = \angle M \), \( \angle KLJ = \angle MLN \) (vertical angles), so AAS, then CPCTC gives \( JK = NM \).

So the key steps:

1. Statement 1: \( L \) is the midpoint of \( \overline{JN} \) (Given)
2. Statement 2: \( \angle KLJ \cong \angle MLN \) (Vertical Angles, Theorem)
3. Statement 3: \( JL = LN \) (Definition of Midpoint)
4. Reason 4: Vertical Angle Theorem (for Statement 2)
5. Statement 5: \( \triangle KLJ \cong \triangle MLN \) (AAS)
6. Reason 6: AAS (Angle - Angle - Side)
7. Statement 7: \( \overline{JK} \cong \overline{NM} \)
8. Reason 8: CPCTC

But since the options for Blank 8 is CPCTC, which is correct for the last step (corresponding parts of congruent triangles are congruent).

Answer:

1. \( L \) is the midpoint of \( \overline{JN} \)
2. \( \angle KLJ \cong \angle MLN \)
3. \( JL = LN \)
4. \( \angle KLJ \cong \angle MLN \)
5. \( \triangle KLJ \cong \triangle MLN \)
6. AAS (Angle - Angle - Side)
7. \( \overline{JK} \cong \overline{NM} \)
8. CPCTC

(Note: The exact filling may depend on the option list, but the geometric proof follows AAS congruence and CPCTC.)