Question

Given: $\overline{\mathrm{AB}}\cong \overline{\mathrm{AC}}\mathbf{;}\mathbf{}\overline{\mathrm{OB}}\mathbf{\cong }\overline{\mathrm{OC}}$

a. Is it possible to prove that $\angle \mathbf{AOB}\cong \angle \mathbf{AOC}$?

b. Is it possible to prove that$\angle \mathbf{AOB}$ and$\angle \mathbf{AOC}$ are right angles?

Short answer

a)Yes, it is possible to prove that $\angle AOB\cong \angle AOC$.

b)No, it is not possible to prove that$\angle AOB$ and$\angle AOC$ are right angles.

Step-by-step solution

Part a Step 1 - Observe the given diagram.

The given diagram is:

Part a Step 2 - Description of step.

It is being given that $\overline{AB}\cong \overline{AC}$and $\overline{OB}\cong \overline{OC}$.

In the triangles $\Delta AOB$and $\Delta AOC$, it can be noticed that:

$$\begin{gathered} \overline{AB}\cong \overline{AC}\left(given\right) \\ \overline{OB}\cong \overline{OC}\left(given\right) \\ \overline{AO}\cong \overline{AO}\left(reflexiveproperty\right) \end{gathered}$$

Therefore, the triangles$\Delta AOB$ and$\Delta AOC$are the congruent triangles by using $SSS$postulate.

Therefore, by using the corresponding parts of congruent triangles it can be said that $\angle AOB\cong \angle AOC$.

Part a Step 3 - Write the conclusion.

Yes, it is possible to prove that $\angle AOB\cong \angle AOC$.

Part b Step 1 - Observe the given diagram.

The given diagram is:

Part b Step 2 - Description of step.

It is being given that $\overline{AB}\cong \overline{AC}$and $\overline{OB}\cong \overline{OC}$.

In the triangles $\Delta AOB$and role="math" localid="1648884343636" $\Delta AOC$, it can be noticed that:

$$\begin{gathered} \overline{AB}\cong \overline{AC}\left(given\right) \\ \overline{OB}\cong \overline{OC}\left(given\right) \\ \overline{AO}\cong \overline{AO}\left(reflexiveproperty\right) \end{gathered}$$

Therefore, the triangles$\Delta AOB$ and$\Delta AOC$are the congruent triangles by using $SSS$postulate.

Therefore, by using the corresponding parts of congruent triangles it can be said that $\angle AOB\cong \angle AOC$.

Part b Step 3 - Write the conclusion.

Therefore, the data is insufficient to prove that$\angle AOB$ and$\angle AOC$ are right angles.

No, it is not possible to prove that $\angle AOB$and $\angle AOC$are right angles.