Question

given: \\( \overline{rs} \cong \overline{ts} \\) and \\( \overline{sb} \\) is an angle bisector.
prove: \\( \overline{rb} \cong \overline{tb} \\)
proof:
which of the following pairs of reasons completes the proof?
○a. definition of congruence
sas criterion
○b. reflexive property
sas criterion
○c. definition of congruence
ssa criterion
○d. reflexive property
ssa criterion

Explanation:

Step1: Analyze the reason for $\overline{SB}\cong\overline{SB}$

Any segment is congruent to itself. This is the reflexive property of congruence. So the reason for $\overline{SB}\cong\overline{SB}$ is the Reflexive Property.

Step2: Determine the triangle - congruence criterion

We have $\overline{RS}\cong\overline{TS}$ (given), $\angle RSB\cong\angle TSB$ (definition of angle bisector), and $\overline{SB}\cong\overline{SB}$ (reflexive property). The Side - Angle - Side (SAS) criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, in $\triangle RSB$ and $\triangle TSB$, we have two sides ($\overline{RS}\cong\overline{TS}$ and $\overline{SB}\cong\overline{SB}$) and the included angle ($\angle RSB\cong\angle TSB$) congruent. So the criterion for $\triangle RSB\cong\triangle TSB$ is the SAS criterion.

Answer:

B. Reflexive Property, SAS Criterion