Question

given: $overline{ao}congoverline{co},overline{ab}congoverline{cb}$
prove: $overline{bo}$ bisects $angle abc$.

Explanation:

Step1: State given information

Given $\overline{AO}\cong\overline{CO},\overline{AB}\cong\overline{CB}$

Step2: Identify reflexive property

$\overline{BO}$ is congruent to itself by reflexive property, $\overline{BO}\cong\overline{BO}$

Step3: Prove triangle congruence

Use SSS postulate. Three pairs of corresponding sides are congruent: $\overline{AO}\cong\overline{CO},\overline{AB}\cong\overline{CB},\overline{BO}\cong\overline{BO}$, so $\triangle ABO\cong\triangle CBO$

Step4: Use CPCTC

Since $\triangle ABO\cong\triangle CBO$, corresponding angles $\angle ABO$ and $\angle CBO$ are congruent

Step5: Apply angle - bisector definition

As $\angle ABO\cong\angle CBO$, $\overline{BO}$ is the angle - bisector of $\angle ABC$

Answer:

1. $\overline{AO}\cong\overline{CO},\overline{AB}\cong\overline{CB}$ - Given
2. $\overline{BO}\cong\overline{BO}$ - Reflexive Property
3. $\triangle ABO\cong\triangle CBO$ - SSS (Side - Side - Side) Congruence Postulate (Since we have $\overline{AO}\cong\overline{CO},\overline{AB}\cong\overline{CB}$ and $\overline{BO}\cong\overline{BO}$)
4. $\angle ABO\cong\angle CBO$ - Corresponding parts of congruent triangles are congruent (CPCTC)
5. $\overline{BO}$ bisects $\angle ABC$ - Definition of an angle - bisector (An angle - bisector divides an angle into two congruent angles)