Question

The proof that ΔRST ≅ ΔVST is shown.

Given: ST is the perpendicular bisector of RV.
Prove: ΔRST ≅ ΔVST

Triangle R S V is cut by perpendicular bisector S T. Point T is the midpoint of line segment R V.

What is the missing reason in the proof?

Statements Reasons
1. ST is the perpendicular bisector of RV. 1. given
2. ∠STR and ∠STV are right angles. 2. def. of perpendicular bisector
3. RS ≅ VS 3. ?
4. ST ≅ ST 4. reflexive property
5. ΔRST ≅ ΔVST 5. HL theorem
perpendicular bisector theorem
converse of the perpendicular bisector theorem
Pythagorean theorem
SSS congruence theor

Answer 1

To find the missing reason for statement 3, "RS ≅ VS", we need to consider the properties of the perpendicular bisector.

Since ST is the perpendicular bisector of segment RV, by the definition of a perpendicular bisector, we know that it divides RV into two equal segments at point T, which is the midpoint of RV. This means that:

\[ RT = TV \]

According to the Perpendicular Bisector Theorem, for any point on the perpendicular bisector of a segment, the distances from that point to the endpoints of the segment are equal. Therefore, we can conclude that:

\[ RS = VS \]

Thus, the missing reason for statement 3 is:

**Perpendicular bisector theorem.**

So the completed proof statements would look like this:

Statements Reasons
1. ST is the perpendicular bisector of RV. 1. given
2. ∠STR and ∠STV are right angles. 2. def. of perpendicular bisector
3. RS ≅ VS 3. perpendicular bisector theorem
4. ST ≅ ST 4. reflexive property
5. ΔRST ≅ ΔVST 5. HL theorem