Question

the proof that △rst ≅ △vst is shown.
given: st is the perpendicular bisector of rv.
prove △rst ≅ △vst
what is the missing reason in the proof?

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} cong overline{vs}) 3. ?
4. (overline{st} cong overline{st}) 4 reflexive property
5. (△rst cong △vst) 5. hl theorem

converse of the perpendicular bisector theorem
pythagorean 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 \( \overline{ST} \) is the perpendicular bisector of \( \overline{RV} \), and \( S \) is a point on \( \overline{ST} \), by the perpendicular bisector theorem, \( \overline{RS}\cong\overline{VS} \). The Pythagorean theorem is about the relationship between the sides of a right - triangle (\(a^{2}+b^{2}=c^{2}\)) and is not relevant here. The converse of the perpendicular bisector theorem would be used to prove that a point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment, which is the opposite of what we need in this proof step.

Answer:

perpendicular bisector theorem