Question

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

Explanation:

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Here, since \(ST\) is the perpendicular bisector of \(RV\) and \(S\) is a point on \(ST\), by the perpendicular bisector theorem, \(RS\cong VS\).
The converse of the perpendicular bisector theorem is about a point equidistant from endpoints implying it is on the perpendicular bisector (not relevant here). The Pythagorean theorem relates sides of a right - triangle (\(a^{2}+b^{2}=c^{2}\), not used for proving segment congruence in this context). The SSS (Side - Side - Side) congruence theorem is for proving triangle congruence (used in step 5 but not for step 3).

Answer:

perpendicular bisector theorem