Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.) answer

$overline{tu}$ is a segment bisector.
$overline{tu}$ is a perpendicular bisector.
$u$ is the vertex of two angles that are congruent to one another.
$u$ is the vertex of a right angle.
$t$ is the midpoint of a segment in the diagram.
$u$ is the midpoint of a segment in the diagram.

Explanation:

Step1: Analyze segment - bisector property

There is no indication that $\overline{TU}$ divides any segment into two equal - length parts, so $\overline{TU}$ is not a segment bisector.

Step2: Analyze perpendicular - bisector property

There is no right - angle symbol or information suggesting that $\overline{TU}$ is perpendicular to and bisects a segment, so $\overline{TU}$ is not a perpendicular bisector.

Step3: Analyze angle - congruence at point U

Since $\overline{RU}\cong\overline{UQ}$, point $U$ is the vertex of $\angle{RUQ}$ and by the isosceles - triangle property (in $\triangle{RUQ}$), the base angles are congruent. So $U$ is the vertex of two angles that are congruent to one another.

Step4: Analyze right - angle at point U

There is no right - angle symbol at $U$, so we cannot say $U$ is the vertex of a right - angle.

Step5: Analyze mid - point property of T

Since $\overline{RT}\cong\overline{TS}$, $T$ is the mid - point of $\overline{RS}$.

Step6: Analyze mid - point property of U

There is no information to suggest that $U$ is the mid - point of a segment.

Answer:

$U$ is the vertex of two angles that are congruent to one another.
$T$ is the midpoint of a segment in the diagram.