Question

select the correct answer from each drop - down menu. given: $overleftrightarrow{mn}$ is the perpendicular bisector of $overline{ab}$. prove: $overline{am} cong overline{bm}$ complete the proof. there is a diagram of a triangle with points a, b, m, n, where n is on ab and m is connected to a, b, n. then there is a two - column proof table with statements and reasons. statements: $overleftrightarrow{mn}$ is the perpendicular bisector of $overline{ab}$. n is the midpoint of $overline{ab}$. drop - down, $angle anm$ and $angle bnm$ are drop - down, drop - down, $overline{mn} cong overline{mn}$, $\triangle anm cong \triangle$ drop - down, $overline{am} cong overline{bm}$. reasons: given, definition of a perpendicular bisector, drop - down, definition of a perpendicular bisector, all right angles are congruent, reflexive property of congruence, sas, cpctc.

Explanation:

Step1: Analyze the first blank (reason for N is the midpoint of $\overline{AB}$)

Since $\overleftrightarrow{MN}$ is the perpendicular bisector of $\overline{AB}$, by the definition of a perpendicular bisector, it bisects $\overline{AB}$, so N is the midpoint. The reason is "definition of a perpendicular bisector".

Step2: Analyze the angle congruence (∠ANM and ∠BNM)

Since $\overleftrightarrow{MN}$ is the perpendicular bisector, ∠ANM and ∠BNM are right angles. All right angles are congruent, so the reason is "All right angles are congruent".

Step3: Analyze the triangle congruence (△ANM ≅ △)

We have AN = BN (N is midpoint), ∠ANM = ∠BNM (right angles), and MN = MN (reflexive). So by SAS, △ANM ≅ △BNM.

Step4: Analyze the final congruence ($\overline{AM}$ ≅ $\overline{BM}$)

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), since △ANM ≅ △BNM, $\overline{AM}$ ≅ $\overline{BM}$.

Answer:

• First blank (reason for N is midpoint): definition of a perpendicular bisector
• Second blank (∠ANM and ∠BNM reason): All right angles are congruent
• Third blank (triangle congruence): △BNM
• Fourth blank (final congruence reason): CPCTC (and the congruence $\overline{AM}$ ≅ $\overline{BM}$ holds)