Question

given: $overline{bx}$ is the perpendicular bisector of $overline{ac}$. prove: $overline{ab}congoverline{bc}$ 1. given 2. definition of a perpendicular bisector 3. $angle bxc$ and $angle bxa$ are right angles. 4. $angle bxccongangle bxa$ 4. all right angles are congruent.

Explanation:

Step1: State given information

Given $\overline{BX}$ is perpendicular bisector of $\overline{AC}$

Step2: Use definition of perpendicular bisector

$\overline{AX}\cong\overline{XC}$ (bisects $\overline{AC}$) and $\overline{BX}\perp\overline{AC}$

Step3: Define right - angle formation

Since $\overline{BX}\perp\overline{AC}$, $\angle BXC$ and $\angle BXA$ are right angles by definition of perpendicular lines

Step4: Apply right - angle congruence

All right angles are congruent, so $\angle BXC\cong\angle BXA$

Step5: Identify reflexive side

$\overline{BX}$ is common to both $\triangle BXA$ and $\triangle BXC$, so $\overline{BX}\cong\overline{BX}$ by reflexive property

Step6: Prove triangle congruence

With $\overline{AX}\cong\overline{XC}$, $\angle BXC\cong\angle BXA$ and $\overline{BX}\cong\overline{BX}$, $\triangle BXA\cong\triangle BXC$ by SAS

Step7: Prove line - segment congruence

Since $\triangle BXA\cong\triangle BXC$, $\overline{AB}\cong\overline{BC}$ by CPCTC

Answer:

1. Given
2. $\overline{AX}\cong\overline{XC}$, $\overline{BX}\perp\overline{AC}$
3. Definition of perpendicular lines
4. All right angles are congruent
5. $\overline{BX}\cong\overline{BX}$ (Reflexive Property)
6. $\triangle BXA\cong\triangle BXC$ (Side - Angle - Side (SAS) congruence criterion)
7. $\overline{AB}\cong\overline{BC}$ (Corresponding parts of congruent triangles are congruent (CPCTC))