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[tex]$\pi$[/tex] is the perpendicular bisector of [tex]$\triangle STU$[/tex], and [tex]$\overline{TX}$[/tex] is a shared side of [tex]$\triangle STX$[/tex] and [tex]$\triangle UTX$[/tex]. Which of the following must be congruent to verify that [tex]$\triangle STX \cong \triangle UTX$[/tex]?

A. [tex]$\overline{ST} \cong \overline{TX}$[/tex]

B. [tex]$\overline{ST} \cong \overline{UX}$[/tex]

C. [tex]$\overline{SX} \cong \overline{TX}$[/tex]

D. [tex]$\overline{SX} \cong \overline{UX}$[/tex]

Sagot :

GinnyAnswer
To determine which pair of line segments must be congruent in order to verify that the triangles [tex]\( \triangle S T X \)[/tex] and [tex]\( \triangle U T X \)[/tex] are congruent, let's analyze the information provided and the congruence criteria.

Given that [tex]\(\pi\)[/tex] is the perpendicular bisector of [tex]\(\triangle S T U\)[/tex] and [tex]\(\overline{T X}\)[/tex] is a shared side of both [tex]\(\triangle S T X\)[/tex] and [tex]\(\triangle U T X\)[/tex], we need to determine which additional congruence condition will confirm the congruence of the two triangles.

1. Shared Side: Both triangles share the side [tex]\(\overline{T X}\)[/tex].

2. Perpendicular Bisector: The perpendicular bisector [tex]\(\pi\)[/tex] implies that [tex]\(\pi\)[/tex] is perpendicular to [tex]\(\overline{S U}\)[/tex] at some point on [tex]\(\overline{S U}\)[/tex], and [tex]\(\pi\)[/tex] splits [tex]\(\overline{S U}\)[/tex] into two equal segments. This means that [tex]\(\pi\)[/tex] makes right angles with [tex]\(\overline{S U}\)[/tex], and segments from [tex]\(S\)[/tex] and [tex]\(U\)[/tex] to the point where [tex]\(\pi\)[/tex] intersects [tex]\(\overline{S U}\)[/tex] are equal. However, for congruence of [tex]\(\triangle S T X\)[/tex] and [tex]\(\triangle U T X\)[/tex], let's identify the correct congruent sides.

3. Side-Angle-Side (SAS) Criterion: To use the SAS criterion for proving the triangles congruent, we need two sides and the included angle to be congruent.

From the given conditions:
- We already have one side [tex]\(\overline{T X}\)[/tex] that is shared.
- Since [tex]\(\pi\)[/tex] is the perpendicular bisector, it makes right angles with [tex]\(\overline{S U}\)[/tex] i.e., [tex]\(\angle S T X\)[/tex] and [tex]\(\angle U T X\)[/tex] are right angles.
- The crucial piece of information here is that the lengths from [tex]\(S\)[/tex] to the point on bisector [tex]\(\pi\)[/tex] must be congruent to the length from [tex]\(U\)[/tex] to the same point. This indicates that [tex]\(\overline{S X} \cong \overline{U X}\)[/tex].

Thus, for the triangles [tex]\(\triangle S T X\)[/tex] and [tex]\(\triangle U T X\)[/tex] to be congruent through SAS criterion, the correct condition that must be met is:

D. [tex]\(\overline{S X} \cong \overline{U X}\)[/tex]

Therefore, choice D is the correct answer.
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