Question

given: $overleftrightarrow{mn}$ is the perpendicular bisector of $overline{ab}$. prove: $overline{am} cong overline{bm}$ complete the proof. statements $overleftrightarrow{mn}$ is the perpendicular bisector of $overline{ab}$. n is the midpoint of $overline{ab}$. $angle anm$ and $angle bnm$ are $overline{mn} cong overline{mn}$ $\triangle anm cong \triangle square$ $overline{am} cong overline{bm}$ reasons given definition of a perpendicular bisector definition of a midpoint definition of a perpendicular bisector all right angles are congruent. reflexive property of congruence sas cpctc

Explanation:

Step1: Analyze the first missing statement

The reason for the next step is "definition of a midpoint", so the statement should be related to the midpoint. Since \( \overleftrightarrow{MN} \) is the perpendicular bisector of \( \overline{AB} \), the midpoint of \( \overline{AB} \) is \( N \), so \( \overline{AN} \cong \overline{BN} \) (by definition of midpoint: a midpoint divides a segment into two congruent segments). Wait, no, looking at the table, the row with reason "definition of a midpoint" should have a statement about midpoint. Wait, the first missing statement (after "N is the midpoint of \( \overline{AB} \)")? Wait, no, let's look at the table structure:

Statements column:
1. \( \overleftrightarrow{MN} \) is the perpendicular bisector of \( \overline{AB} \) (given)
2. N is the midpoint of \( \overline{AB} \) (reason: definition of a perpendicular bisector? Wait no, the reason for "N is the midpoint of \( \overline{AB} \)" should be "definition of a perpendicular bisector" because a perpendicular bisector intersects the segment at its midpoint. Wait, the reason for that row is "definition of a perpendicular bisector"? Wait the reasons are in the bottom column. Let's list the rows:

Row 1: Statement: \( \overleftrightarrow{MN} \) is the perpendicular bisector of \( \overline{AB} \); Reason: given

Row 2: Statement: N is the midpoint of \( \overline{AB} \); Reason: definition of a perpendicular bisector (because a perpendicular bisector of a segment passes through its midpoint)

Row 3: Statement: \( \angle ANM \) and \( \angle BNM \) are...; Reason: definition of a perpendicular bisector (because a perpendicular bisector is perpendicular to the segment, so those angles are right angles)

Row 4: Statement: (something); Reason: All right angles are congruent. So \( \angle ANM \cong \angle BNM \) (since they are right angles)

Row 5: Statement: \( \overline{MN} \cong \overline{MN} \); Reason: reflexive property of congruence

Row 6: Statement: \( \triangle ANM \cong \triangle BNM \); Reason: SAS (since \( \overline{AN} \cong \overline{BN} \) (midpoint), \( \angle ANM \cong \angle BNM \) (right angles), \( \overline{MN} \cong \overline{MN} \) (reflexive))

Row 7: Statement: \( \overline{AM} \cong \overline{BM} \); Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Wait, let's fill in the missing statements:

1. First missing statement (after "N is the midpoint of \( \overline{AB} \)"): Wait no, the first missing statement is in the row with reason "definition of a midpoint"? Wait no, the reason for "N is the midpoint of \( \overline{AB} \)" should be "definition of a perpendicular bisector" (because a perpendicular bisector intersects the segment at its midpoint). Then the next row: " \( \angle ANM \) and \( \angle BNM \) are right angles" (reason: definition of a perpendicular bisector, because a perpendicular bisector is perpendicular to the segment, so those angles are 90 degrees, i.e., right angles). Then the next row: " \( \angle ANM \cong \angle BNM \)" (reason: All right angles are congruent). Then " \( \overline{MN} \cong \overline{MN} \)" (reflexive), then " \( \triangle ANM \cong \triangle BNM \)" (SAS: \( \overline{AN} \cong \overline{BN} \) (midpoint), \( \angle ANM \cong \angle BNM \), \( \overline{MN} \cong \overline{MN} \)), then " \( \overline{AM} \cong \overline{BM} \)" (CPCTC).

Wait, let's correct:

- Row with reason "definition of a midpoint": Wait no, the reason "definition of a midpoint" should correspond to a statement about midpoint, like \( \overline{AN} \cong \overline{BN} \) (since N is the midpoint, so AN = BN). But in the table, the row after "N is the midpoint of \( \overline{AB} \)" has reason "definition of a midpoint"? Wait the table's reasons column:

Reasons:

1. given

2. definition of a perpendicular bisector? Wait no, the second row (N is the midpoint of \( \overline{AB} \)) has reason "definition of a midpoint"? No, the reason for "N is the midpoint of \( \overline{AB} \)" is "definition of a perpendicular bisector" because a perpendicular bisector must pass through the midpoint. Then the next row ( \( \angle ANM \) and \( \angle BNM \) are...) has reason "definition of a perpendicular bisector" (because a perpendicular bisector is perpendicular to the segment, so those angles are right angles). Then the next row (angle congruence) has reason "All right angles are congruent". Then \( \overline{MN} \cong \overline{MN} \) (reflexive), then \( \triangle ANM \cong \triangle BNM \) (SAS), then \( \overline{AM} \cong \overline{BM} \) (CPCTC).

Wait, let's fill the missing statements:

1. After "N is the midpoint of \( \overline{AB} \)", the next statement (with reason "definition of a midpoint")? No, the reason "definition of a midpoint" should be for \( \overline{AN} \cong \overline{BN} \). So:

- Statement: \( \overline{AN} \cong \overline{BN} \); Reason: definition of a midpoint (because N is the midpoint, so AN = BN)

- Then, \( \angle ANM \) and \( \angle BNM \) are right angles; Reason: definition of a perpendicular bisector (because \( \overleftrightarrow{MN} \) is perpendicular to \( \overline{AB} \), so those angles are 90°)

- Then, \( \angle ANM \cong \angle BNM \); Reason: All right angles are congruent

- Then, \( \overline{MN} \cong \overline{MN} \); Reason: reflexive property of congruence

- Then, \( \triangle ANM \cong \triangle BNM \); Reason: SAS ( \( \overline{AN} \cong \overline{BN} \), \( \angle ANM \cong \angle BNM \), \( \overline{MN} \cong \overline{MN} \))

- Then, \( \overline{AM} \cong \overline{BM} \); Reason: CPCTC

But looking at the table, the first missing statement (the empty box after "N is the midpoint of \( \overline{AB} \)") should be \( \overline{AN} \cong \overline{BN} \) (reason: definition of a midpoint). Then the next empty box (for \( \angle ANM \) and \( \angle BNM \)) should be "right angles" (reason: definition of a perpendicular bisector). Then the next empty box (angle congruence) is \( \angle ANM \cong \angle BNM \) (reason: All right angles are congruent). Then \( \overline{MN} \cong \overline{MN} \) (reflexive), then \( \triangle ANM \cong \triangle BNM \) (SAS), then \( \overline{AM} \cong \overline{BM} \) (CPCTC).

Wait, the table's statements column:

1. \( \overleftrightarrow{MN} \) is the perpendicular bisector of \( \overline{AB} \)

2. N is the midpoint of \( \overline{AB} \)

3. [Empty] (reason: definition of a midpoint) → \( \overline{AN} \cong \overline{BN} \)

4. \( \angle ANM \) and \( \angle BNM \) are [Empty] (reason: definition of a perpendicular bisector) → right angles

5. [Empty] (reason: All right angles are congruent) → \( \angle ANM \cong \angle BNM \)

6. \( \overline{MN} \cong \overline{MN} \)

7. \( \triangle ANM \cong \triangle BNM \)

8. \( \overline{AM} \cong \overline{BM} \)

But the user's table has:

Statements:

- \( \overleftrightarrow{MN} \) is the perpendicular bisector of \( \overline{AB} \)

- N is the midpoint of \( \overline{AB} \)

- [Empty] (reason: definition of a midpoint)

- \( \angle ANM \) and \( \angle BNM \) are [Empty] (reason: definition of a perpendicular bisector)

- [Empty] (reason: All right angles are congruent)

- \( \overline{MN} \cong \overline{MN} \)

- \( \triangle ANM \cong \triangle BNM \)

- \( \overline{AM} \cong \overline{BM} \)

Reasons:

- given

- definition of a perpendicular bisector? Wait no, the reason for "N is the midpoint of \( \overline{AB} \)" is "definition of a perpendicular bisector" (because a perpendicular bisector intersects the segment at its midpoint). Then the next row (reason: definition of a midpoint) should have statement \( \overline{AN} \cong \overline{BN} \) (since N is the midpoint, AN = BN). Then the next row (reason: definition of a perpendicular bisector) should have statement " \( \angle ANM \) and \( \angle BNM \) are right angles" (because a perpendicular bisector is perpendicular to the segment, so those angles are 90°). Then the next row (reason: All right angles are congruent) should have statement " \( \angle ANM \cong \angle BNM \)".

So filling in:

1. The first empty statement (reason: definition of a midpoint): \( \overline{AN} \cong \overline{BN} \)

2. The second empty statement ( \( \angle ANM \) and \( \angle BNM \) are...): right angles

3. The third empty statement (reason: All right angles are congruent): \( \angle ANM \cong \angle BNM \)

Then, the triangle congruence: \( \triangle ANM \cong \triangle BNM \) by SAS (since \( \overline{AN} \cong \overline{BN} \), \( \angle ANM \cong \angle BNM \), \( \overline{MN} \cong \overline{MN} \)), and then \( \overline{AM} \cong \overline{BM} \) by CPCTC.

But the user's table has " \( \triangle ANM \cong \triangle \) [Empty]" (the triangle name). Since \( \triangle ANM \) and \( \triangle BNM \) are congruent, so \( \triangle ANM \cong \triangle BNM \).

So putting it all together:

- For the row with reason "definition of a midpoint": Statement is \( \overline{AN} \cong \overline{BN} \)

- For the row with \( \angle ANM \) and \( \angle BNM \) are...: Statement is "right angles" (so \( \angle ANM \) and \( \angle BNM \) are right angles)

- For the row with reason "All right angles are congruent": Statement is \( \angle ANM \cong \angle BNM \)

- For the triangle congruence: \( \triangle ANM \cong \triangle BNM \)

Step2: Verify the triangle congruence

To prove \( \overline{AM} \cong \overline{BM} \), we use SAS congruence for triangles \( ANM \) and \( BNM \):

- \( \overline{AN} \cong \overline{BN} \) (midpoint, so AN = BN)

- \( \angle ANM \cong \angle BNM \) (both right angles, so congruent)

- \( \overline{MN} \cong \overline{MN} \) (reflexive property)

Thus, \( \triangle ANM \cong \triangle BNM \) by SAS. Then, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \overline{AM} \cong \overline{BM} \).

Answer:

The missing statements are:

1. \( \overline{AN} \cong \overline{BN} \) (reason: definition of a midpoint)

1. \( \angle ANM \) and \( \angle BNM \) are right angles (reason: definition of a perpendicular bisector)

1. \( \angle ANM \cong \angle BNM \) (reason: All right angles are congruent)

1. \( \triangle ANM \cong \triangle BNM \) (reason: SAS)

And finally, \( \overline{AM} \cong \overline{BM} \) by CPCTC.

(Note: Depending on the exact empty boxes, the key completions are \( \overline{AN} \cong \overline{BN} \), "right angles", \( \angle ANM \cong \angle BNM \), and \( \triangle ANM \cong \triangle BNM \))