Question

the proof that δrst ≅ δvst is shown. given: $overline{st}$ is the perpendicular bisector of $overline{rv}$. prove: δrst ≅ δvst what is the missing reason in the proof? statements reasons 1. $overline{st}$ is the perpendicular bisector of $overline{rv}$. 1. given 2. ∠str and ∠stv are right angles. 2. def of perpendicular bisector 3. $overline{rs}$ ≅ $overline{vs}$ 3. ? 4. $overline{st}$ ≅ $overline{st}$ 4. reflexive property 5. δrst ≅ δvst 5. hl theorem options: perpendicular bisector theorem, converse of the perpendicular bisector theorem, pythagorean theorem, sss congruence theorem

Explanation:

To determine the missing reason for \(\overline{RS}\cong\overline{VS}\), we recall the perpendicular bisector theorem, which states that any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints. Since \(S\) lies on the perpendicular bisector of \(\overline{RV}\) (given that \(\overline{ST}\) is the perpendicular bisector of \(\overline{RV}\)), \(S\) must be equidistant from \(R\) and \(V\), so \(\overline{RS}\cong\overline{VS}\). The other options do not apply: the converse of the perpendicular bisector theorem relates to a point being on the bisector if equidistant, the Pythagorean theorem deals with right - triangle side lengths, and the SSS congruence theorem is for triangle congruence (not for this segment congruence step).

Answer:

perpendicular bisector theorem