Question

in the diagram to the left, segments $overline{qr},overline{st}$, and $overline{uv}$ congruent to each other. segments $overline{rs}$ and $overline{tu}$ are congruent to each other. use the segment addition postulate to determine what other segments must be congruent. (select all that apply.) $overline{qu}$ and $overline{rv}$ must be congruent $overline{ru}$ and $overline{tv}$ must be congruent $overline{qt}$ and $overline{sv}$ must be congruent $overline{ru}$ and $overline{qv}$ must be congruent $overline{qs},overline{rt},overline{su}$, and $overline{tv}$ must be congruent $overline{qt}$ and $overline{rv}$ must be congruent

Explanation:

Step1: Recall Segment - Addition Postulate

If \(AB = CD\) and \(BC=DE\), then \(AC = CE\). Given \(\overline{QR}\cong\overline{ST}\cong\overline{UV}\) and \(\overline{RS}\cong\overline{TU}\).

Step2: Analyze option 1

\(\overline{QU}=\overline{QR}+\overline{RS}+\overline{ST}+\overline{TU}\) and \(\overline{RV}=\overline{RS}+\overline{ST}+\overline{TU}+\overline{UV}\). Since \(\overline{QR}\cong\overline{UV}\), by the Segment - Addition Postulate, \(\overline{QU}\cong\overline{RV}\).

Step3: Analyze option 2

\(\overline{RU}=\overline{RS}+\overline{ST}+\overline{TU}\) and \(\overline{TV}=\overline{ST}+\overline{TU}+\overline{UV}\). Just because \(\overline{RS}\cong\overline{TU}\) and \(\overline{QR}\cong\overline{ST}\cong\overline{UV}\) does not mean \(\overline{RU}\cong\overline{TV}\).

Step4: Analyze option 3

\(\overline{QT}=\overline{QR}+\overline{RS}+\overline{ST}\) and \(\overline{SV}=\overline{ST}+\overline{TU}+\overline{UV}\). Since \(\overline{QR}\cong\overline{UV}\) and \(\overline{RS}\cong\overline{TU}\), by the Segment - Addition Postulate, \(\overline{QT}\cong\overline{SV}\).

Step5: Analyze option 4

\(\overline{RU}=\overline{RS}+\overline{ST}+\overline{TU}\) and \(\overline{QV}=\overline{QR}+\overline{RS}+\overline{ST}+\overline{TU}+\overline{UV}\). So \(\overline{RU}\) and \(\overline{QV}\) are not congruent.

Step6: Analyze option 5

There is no information or application of the Segment - Addition Postulate to show that \(\overline{QS}\), \(\overline{RT}\), \(\overline{SU}\), and \(\overline{TV}\) are congruent.

Step7: Analyze option 6

\(\overline{QT}=\overline{QR}+\overline{RS}+\overline{ST}\) and \(\overline{RV}=\overline{RS}+\overline{ST}+\overline{TU}+\overline{UV}\). They are not congruent.

Answer:

A. \(\overline{QU}\) and \(\overline{RV}\) must be congruent
C. \(\overline{QT}\) and \(\overline{SV}\) must be congruent