To determine which sides must be congruent in order to verify that triangle [tex]\( \triangle STX \cong \triangle UTX \)[/tex], let's analyze the given information step-by-step:
1. Perpendicular Bisector:
Given that [tex]\( \overline{TX} \)[/tex] is the perpendicular bisector of [tex]\( \triangle STU \)[/tex], this means [tex]\( TX \)[/tex] intersects [tex]\( ST \)[/tex] and [tex]\( TU \)[/tex] at [tex]\( T \)[/tex], creating a right angle. It also means that [tex]\( T \)[/tex] bisects the segment [tex]\( SU \)[/tex], implying [tex]\( SX = UX \)[/tex].
2. Common Side:
Both triangles [tex]\( \triangle STX \)[/tex] and [tex]\( \triangle UTX \)[/tex] share the side [tex]\( TX \)[/tex].
By analyzing these points:
- Perpendicular Bisector Property: Since [tex]\( \overline{TX} \)[/tex] bisects [tex]\( \overline{SU} \)[/tex] at [tex]\( T \)[/tex], this results in [tex]\( \overline{SX} \cong \overline{UX} \)[/tex].
- Shared Side: The side [tex]\( \overline{TX} \)[/tex] is common to both triangles [tex]\( \triangle STX \)[/tex] and [tex]\( \triangle UTX \)[/tex].
For the triangles [tex]\( \triangle STX \)[/tex] and [tex]\( \triangle UTX \)[/tex] to be congruent ( [tex]\( \triangle STX \cong \triangle UTX \)[/tex] ), the congruent sides required by the given choices must be:
Choice C: [tex]\( \overline{SX} \cong \overline{UX} \)[/tex]
This is because [tex]\( \overline{SX} = \overline{UX} \)[/tex] is derived from the property of the perpendicular bisector.
Thus, the correct answer is:
C. [tex]\( \overline{SX} \cong \overline{UX} \)[/tex]
