complete the paragraph proof.
given: $ \angle abr $ and $ \angle acr $ are right angles
$ \overline{ab} \cong \overline{bc} $
$ \overline{bc} \cong \overline{ac} $
prove: $ ar $ bisects $ \angle bac $
it is given that $ \angle abr $ and $ \angle acr $ are right angles, $ \overline{ab} \cong \overline{bc} $ and
$ \overline{bc} \cong \overline{ac} $. since they contain right angles, $ \triangle abr $ and $ \triangle acr $ are right
riangles. the right triangles share hypotenuse $ \overline{ar} $, and reflexive property
justifies that $ \overline{ar} \cong \overline{ar} $. since $ \overline{ab} \cong \overline{bc} $ and $ \overline{bc} \cong \overline{ac} $, the transitive
property justifies $ \overline{ab} \cong \overline{ac} $. now, the hypotenuse and leg of right $ \triangle abr $ is
congruent to the hypotenuse and the leg of right $ \triangle acr $, so
$ \triangle abr \cong \triangle acr $ by the hl congruence postulate. therefore, $ \text{__________} $ by
cpctc, and $ \overline{ar} $ bisects $ \angle bac $ by the definition of bisector.
$ \angle apb \cong \angle apc $
$ \angle bar \cong \angle car $
$ \angle bpr \cong \angle cpr $
$ \angle apc \cong \angle bpr $

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# Brief Explanations:
CPCTC (Corresponding Parts of Congruent Triangles are Congruent) states that if two triangles are congruent, then their corresponding parts (angles and sides) are congruent. Since we have shown that $\triangle ABR\cong\triangle ACR$ and we want to prove that $AR$ bisects $\angle BAC$, we need the angles $\angle BAR$ and $\angle CAR$ (which are parts of $\angle BAC$) to be congruent. The other angle pairs ($\angle APB\cong\angle APC$, $\angle BPR\cong\angle CPR$, $\angle APC\cong\angle BPR$) are not the angles that make up $\angle BAC$ for the angle - bisector definition.

# Answer:
$\angle BAR\cong\angle CAR$

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