Question

complete the paragraph proof. given: ∠abr and ∠acr are right angles, ( overline{ab} cong overline{bc} ), ( overline{bc} cong overline{ac} ). prove: ( overline{ar} ) bisects ( angle bac ). diagram with points b, a, r, c, p it is given that ∠abr and ∠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 triangles. 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, ______ by cpctc, and ( overline{ar} ) bisects ( angle bac ) by the definition of bisector. options: ( angle bar cong angle car ), ( angle bpr cong angle cpr )

Explanation:

We know that \(\triangle ABR \cong \triangle ACR\) by the HL congruence postulate. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) tells us that corresponding angles of congruent triangles are congruent. The angles \(\angle BAR\) and \(\angle CAR\) are corresponding angles of \(\triangle ABR\) and \(\triangle ACR\) (since \(AR\) is the common hypotenuse and we're looking at the angles at \(A\) formed by \(AR\) with \(AB\) and \(AC\) respectively). The other option, \(\angle BPR \cong \angle CPR\), is not related to the angles of \(\angle BAC\) which we need to show are bisected. So by CPCTC, \(\angle BAR \cong \angle CAR\).

Answer:

\(\boldsymbol{\angle BAR \cong \angle CAR}\)