Question

what additional information is required for the 2 triangles to be congruent by sas? a) $overline{st}congoverline{sc}$ b) $overline{tr}congoverline{ce}$ c) $angle rcongangle e$ or $angle rstcongangle esc$ d) $overline{rs}congoverline{es}$

Explanation:

Step1: Recall SAS congruence criterion

The Side - Angle - Side (SAS) congruence criterion states that two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.

Step2: Analyze the given triangles

In the two triangles $\triangle RST$ and $\triangle ESC$, we need to identify the sides and angles. We already have an angle (the vertical angles $\angle RST$ and $\angle ESC$ are equal). To use SAS, we need to have two pairs of corresponding sides equal. We know that one side - related information is missing.

Step3: Evaluate each option

• Option A: $\overline{ST}\cong\overline{SC}$ gives only one side, not the pair of sides needed for SAS.
• Option B: $\overline{TR}\cong\overline{CE}$ is not a pair of corresponding sides for the two triangles with respect to the included - angle situation for SAS.
• Option C: $\angle R\cong\angle E$ or $\angle RST\cong\angle ESC$ gives angle information. We already have the vertical - angle equality, and we need side information for SAS.
• Option D: $RS\cong ES$ gives one of the required sides. Along with the vertical - angle equality and if we assume we can find or have information about the other pair of sides and included angle, this satisfies the SAS criterion.

Answer:

D. $RS\cong ES$