Question

given: ∠abr and ∠acr are right angles
\\( \overline { a b } \cong \overline { b c } \\)
\\( \overline { b c } \cong \overline { a c } \\)
prove: \\( \overline { a r } \\) bisects \\( \angle b a c \\)
and \\( b c \cong a c \\). since they contain right angles, \\( \triangle a b r \\)
and \\( \triangle a c r \\) are right triangles. the right triangles share
hypotenuse \\( \overline { a r } \\), and reflexive property justifies that
\\( \overline { a r } \cong \overline { a r } \\). since \\( \overline { a b } \cong \overline { b c } \\) and \\( \overline { b c } \cong \overline { a c } \\), the transitive
property justifies \\( \overline { a b } \cong \overline { a c } \\). now, the hypotenuse and leg
of right \\( \triangle a b r \\) is congruent to the hypotenuse and the
leg of right \\( \triangle a c r \\), so \\( \triangle a b r \cong \triangle a c r \\) by the hl
congruence postulate. therefore, \\( \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ \\) by
cpctc, and \\( \overline { a r } \\) bisects \\( \angle b a c \\) by the definition of
bisector.
\\( \bigcirc \angle b a r \cong \angle c a r \\)
\\( \bigcirc \angle b p r \cong \angle c p r \\)
\\( \bigcirc \angle a p c \cong \angle b p r \\)
\\( \bigcirc \angle a p b \cong \angle a p c \\)

Explanation:

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 \(\overline{AR}\) bisects \(\angle BAC\) (i.e., \(\angle BAR=\angle CAR\)), we need the corresponding angles \(\angle BAR\) and \(\angle CAR\) of the congruent triangles \(\triangle ABR\) and \(\triangle ACR\) to be congruent.

Answer:

\(\angle BAR\cong\angle CAR\)